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== 9.2#h(0.6em)Coordinate Form


    
      
=== Introduction


      In Section 9.1 we saw how to resolve a vector into horizontal and vertical components. Thus, every vector can be expressed as the sum of a horizontal vector and a vertical vector.


      #examplebox("Example 1")[][
A vector #math.equation(block: false, alt: "w")[$bold(w)$] has magnitude 4 and direction #math.equation(block: false, alt: "θ equals 29 degrees")[$θ = 29^(∘)$], where #math.equation(block: false, alt: "θ")[$θ$] is measured counter-clockwise from the positive #math.equation(block: false, alt: "x")[$x$]-axis. Express #math.equation(block: false, alt: "w")[$bold(w)$] as the sum of a horizontal vector, #math.equation(block: false, alt: "w sub x")[$bold(w)_(bold(x))$], and a vertical vector, #math.equation(block: false, alt: "w sub y")[$bold(w)_(bold(y))$].


        
        #solutionbox[
The components of #math.equation(block: false, alt: "w")[$bold(w)$] are given by


          #math.equation(block: true, alt: "w sub x, equals ∥ w ∥ cos ⁡ open parenthesis θ close parenthesis; equals 4 cos ⁡ open parenthesis 29 degrees close parenthesis equals 3.50")[$w_(x) & = ∥ bold(w) ∥ cos ( θ ) \  & = 4 cos ( 29^(∘) ) = 3.50$]
          #math.equation(block: true, alt: "w sub y, equals ∥ w ∥ sin ⁡ open parenthesis θ close parenthesis; equals 4 sin ⁡ open parenthesis 29 degrees close parenthesis equals 1.94")[$w_(y) & = ∥ bold(w) ∥ sin ( θ ) \  & = 4 sin ( 29^(∘) ) = 1.94$]
          #figure(figph[vectors on coordinate plane], alt: "vectors on coordinate plane", caption: none)


          Then #math.equation(block: false, alt: "w equals w sub x plus w sub y")[$" " bold(w) = bold(w)_(bold(x)) + bold(w)_(bold(y))$], where #math.equation(block: false, alt: "w sub x")[$bold(w)_(bold(x))$] is a horizontal vector of magnitude 3.50, and #math.equation(block: false, alt: "w sub y")[$bold(w)_(bold(y))$] is a vertical vector of magnitude 1.94. See the figure above.
]
]


      
        A vector #math.equation(block: false, alt: "u")[$bold(u)$] has magnitude 2 and direction #math.equation(block: false, alt: "θ equals 116 degrees")[$θ = 116^(∘)$], where #math.equation(block: false, alt: "θ")[$θ$] is in standard position. Express #math.equation(block: false, alt: "u")[$bold(u)$] as the sum of a horizontal vector, #math.equation(block: false, alt: "u sub x")[$bold(u)_(bold(x))$], and a vertical vector, #math.equation(block: false, alt: "u sub y")[$bold(u)_(bold(y))$].


        
          #math.equation(block: false, alt: "u equals u sub x plus u sub y")[$bold(u) = bold(u)_(bold(x)) + bold(u)_(bold(y))$], where #math.equation(block: false, alt: "∥ u sub x ∥ equals 1.941")[$" " ∥ bold(u)_(bold(x)) ∥ = 1.941 " "$] and #math.equation(block: false, alt: "u sub x")[$bold(u)_(bold(x))$] points to the left; #math.equation(block: false, alt: "∥ u sub y ∥ equals 0.484")[$" " ∥ bold(u)_(bold(y)) ∥ = 0.484 " "$] and #math.equation(block: false, alt: "u sub y")[$bold(u)_(bold(y))$] points upward.


        
      
    
    
      
=== Unit Vectors


      It is often useful to describe a vector by giving its horizontal and vertical components, instead of its magnitude and direction. To make the notation easier, we give names to the vectors of length 1 that point in the #math.equation(block: false, alt: "x")[$x$]- and #math.equation(block: false, alt: "y")[$y$]-directions. A vector of magnitude 1 is called a #strong[unit vector]. We can have unit vectors in any direction, but the unit vector in the #math.equation(block: false, alt: "x")[$x$]-direction is denoted by #math.equation(block: false, alt: "i")[$bold(i)$], and the unit vector in the #math.equation(block: false, alt: "y")[$y$]-direction is called #math.equation(block: false, alt: "j")[$bold(j)$], as shown below.


      By taking scalar multiples of #math.equation(block: false, alt: "i")[$bold(i)$] and #math.equation(block: false, alt: "j")[$bold(j)$], we can describe any vector that lies in the directions of the coordinate axes. For example, #math.equation(block: false, alt: "4 i")[$4 bold(i)$] represents the vector of magnitude 4 pointing in the #math.equation(block: false, alt: "x")[$x$]-direction, and #math.equation(block: false, alt: "3 j")[$3 bold(j)$] represents the vector of magnitude 3 pointing in the#math.equation(block: false, alt: "y")[$y$]-direction. And by adding multiples of #math.equation(block: false, alt: "i")[$bold(i)$] and #math.equation(block: false, alt: "j")[$bold(j)$], we can represent any vector we like. The vector #math.equation(block: false, alt: "v equals 4 i plus 3 j")[$bold(v) = 4 bold(i) + 3 bold(j)$] is shown at right.


      #figure(figph[vectors on xy-plane], alt: "vectors on xy-plane", caption: none)


      Because the components of the vector are chosen to align with the coordinate system, we call this the #strong[coordinate form] of the vector.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Coordinate Form of a Vector]


        The vector


        #math.equation(block: true, alt: "v equals a i plus b j")[$bold(v) = a bold(i) + b bold(j)$]
        is the vector whose horizontal component is #math.equation(block: false, alt: "a")[$a$] and whose vertical component is #math.equation(block: false, alt: "b")[$b$].
]


      The coordinate form of the vector #math.equation(block: false, alt: "w")[$bold(w)$] in the previous example is #math.equation(block: false, alt: "w equals 3.50 i plus 1.94 j")[$bold(w) = 3.50 bold(i) + 1.94 bold(j)$].


      #examplebox("Example 2")[][
State the coordinate form of the vector shown at right.


          #figure(figph[vectors on xy-plane], alt: "vectors on xy-plane", caption: none)


        
        #solutionbox[
From its base to its tip, the vector extends 4 units in the negative #math.equation(block: false, alt: "i")[$bold(i)$] direction and 6 units in the #math.equation(block: false, alt: "j")[$bold(j)$] direction. Thus, #math.equation(block: false, alt: "v equals minus 4 i plus 6 j")[$bold(v) = − 4 bold(i) + 6 bold(j)$].
]
]


      
        State the coordinate form of the vector shown at right.


        #figure(figph[vectors], alt: "vectors", caption: none)


        
          #math.equation(block: true, alt: "w equals 3 i minus 5 j")[$bold(w) = 3 bold(i) − 5 bold(j)$]


        
      
    
    
      
=== Converting Between Geometric and Coordinate Form


      It is a simple matter to find the magnitude and direction of a vector given in coordinate form. The vector #math.equation(block: false, alt: "v equals 4 i plus 3 j")[$bold(v) = 4 bold(i) + 3 bold(j)$] has magnitude


      #math.equation(block: true, alt: "∥ v ∥ equals the square root of 3 squared plus 4 squared equals the square root of 25 equals 5")[$∥ bold(v) ∥ = sqrt(3^(2) + 4^(2)) = sqrt(25) = 5$]
      and direction #math.equation(block: false, alt: "θ equals tan to the power minus 1 ⁡ open parenthesis the fraction 3 over 4 close parenthesis equals 36.9 degrees")[$θ = tan^(− 1) ( display(frac(3, 4)) ) = 36.9^(∘)$]. Thus, we can readily convert vectors from geometric form to coordinate form or vice versa.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Comparing the Geometric and Coordinate Forms of a Vector]


        Suppose that the vector #math.equation(block: false, alt: "v")[$bold(v)$] has magnitude #math.equation(block: false, alt: "∥ v ∥")[$∥ bold(v) ∥$] and points in the direction of the angle #math.equation(block: false, alt: "θ")[$θ$] in standard position. If #math.equation(block: false, alt: "v")[$bold(v)$] has the coordinate form #math.equation(block: false, alt: "v equals a i plus b j")[$bold(v) = a bold(i) + b bold(j)$], then


        #math.equation(block: true, alt: "a, equals ∥ v ∥ cos ⁡ open parenthesis θ close parenthesis, ∥ v ∥, equals the square root of a squared plus b squared; b, equals ∥ v ∥ sin ⁡ open parenthesis θ close parenthesis, tan ⁡ open parenthesis θ close parenthesis, equals the fraction b over a")[$bold(italic(a)) & " " = ∥ bold(v) ∥ bold(cos) ( bold(italic(θ)) ) &  &  & ∥ bold(v) ∥ & " " = sqrt(bold(italic(a))^(2) + bold(italic(b))^(2)) \ bold(italic(b)) & " " = ∥ bold(v) ∥ bold(sin) ( bold(italic(θ)) ) &  &  & bold(tan) ( bold(italic(θ)) ) & " " = display(frac(bold(italic(b)), bold(italic(a))))$]
]


      #examplebox("Example 3")[][
+ Find the geometric form of the vector #math.equation(block: false, alt: "w equals minus 3 i minus 2 j")[$bold(w) = − 3 bold(i) − 2 bold(j)$].
+ Find the coordinate form of the vector whose magnitude is 5 and whose direction is the angle #math.equation(block: false, alt: "θ equals 135 degrees")[$θ = 135^(∘)$].


        
        #solutionbox[
+ The magnitude of #math.equation(block: false, alt: "w")[$bold(w)$] is #math.equation(block: false, alt: "∥ w ∥ equals the square root of open parenthesis minus 3 close parenthesis squared plus open parenthesis minus 2 close parenthesis squared equals the square root of 13")[$∥ bold(w) ∥ = sqrt(( − 3 )^(2) + ( − 2 )^(2)) = sqrt(13)$]. The direction  satisfies the equation #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis equals the fraction minus 2 over minus 3")[$tan ( θ ) = display(frac(− 2, − 3))$]. Because the point #math.equation(block: false, alt: "open parenthesis minus 3 , minus 2 close parenthesis")[$( − 3 , − 2 )$] lies in the third quadrant, we know that #math.equation(block: false, alt: "θ")[$θ$] must be a third quadrant angle. Thus, #math.equation(block: false, alt: "θ equals tan to the power minus 1 ⁡ open parenthesis the fraction 2 over 3 close parenthesis plus 180 degrees equals 213.69 degrees")[$θ = tan^(− 1) ( display(frac(2, 3)) ) + 180^(∘) = 213.69^(∘)$].
+ The coordinates of the vector are given by 
        #math.equation(block: true, alt: "a equals 5 cos ⁡ open parenthesis 135 degrees close parenthesis equals the fraction minus 5 over the square root of 2 , b equals 5 sin ⁡ open parenthesis 135 degrees close parenthesis equals the fraction 5 over the square root of 2")[$a = 5 cos ( 135^(∘) ) = display(frac(− 5, sqrt(2))) , " " " " " " " " b = 5 sin ( 135^(∘) ) = display(frac(5, sqrt(2)))$]
  so the coordinate form is #math.equation(block: false, alt: "w equals the fraction minus 5 over the square root of 2 i plus the fraction 5 over the square root of 2 j")[$bold(w) = display(frac(− 5, sqrt(2))) bold(i) + display(frac(5, sqrt(2))) bold(j)$].
]
]


      
        Find the geometric form of the vector #math.equation(block: false, alt: "F equals minus 6 i plus 8 j")[$bold(F) = − 6 bold(i) + 8 bold(j)$].


        
          magnitude 10, direction #math.equation(block: false, alt: "127 degrees")[$127^(∘)$]


        
      
    
    
      
=== Scalar Multiples of Vectors in Coordinate Form


      Scalar multiplication is easy to compute in coordinate form. This is really a consequence of the fact that the sides of similar triangles are proportional.


      The figure at right shows the vectors #math.equation(block: false, alt: "u equals 3 i plus 2 j")[$bold(u) = 3 bold(i) + 2 bold(j)$] and #math.equation(block: false, alt: "v equals 2 u")[$bold(v) = 2 bold(u)$]. The vector #math.equation(block: false, alt: "v")[$bold(v)$] is twice as long as #math.equation(block: false, alt: "u")[$bold(u)$] and points in the same direction as #math.equation(block: false, alt: "u")[$bold(u)$]. You can see that each component of #math.equation(block: false, alt: "v")[$bold(v)$] is twice the corresponding component of #math.equation(block: false, alt: "u")[$bold(u)$], so that #math.equation(block: false, alt: "v equals 6 i plus 4 j")[$bold(v) = 6 bold(i) + 4 bold(j)$].


      #figure(figph[vectors], alt: "vectors", caption: none)


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Scalar Multiplication in Coordinate Form]


        If #math.equation(block: false, alt: "v equals a i plus b j")[$bold(v) = a bold(i) + b bold(j)$] and #math.equation(block: false, alt: "k")[$k$] is a scalar, then


        #math.equation(block: true, alt: "k v equals k a i plus k b j")[$bold(italic(k)) bold(v) = bold(italic(k)) bold(italic(a)) bold(i) + bold(italic(k)) bold(italic(b)) bold(j)$]
]


      In other words, to find a scalar multiple of a vector in coordinate form, we multiply each component by the scalar.


      #examplebox("Example 4")[][
The vector #math.equation(block: false, alt: "v")[$bold(v)$] has coordinate form #math.equation(block: false, alt: "v equals 4 i minus 7 j")[$bold(v) = 4 bold(i) − 7 bold(j)$].  Find the coordinate form for the vector #math.equation(block: false, alt: "w equals minus the fraction 1 over 2 v")[$bold(w) = − display(frac(1, 2)) bold(v)$].


        
        #solutionbox[
We multiply each component of #math.equation(block: false, alt: "v")[$bold(v)$] by #math.equation(block: false, alt: "minus the fraction 1 over 2")[$− display(frac(1, 2))$] to get #math.equation(block: false, alt: "w equals minus 2 i plus the fraction 7 over 2 j")[$" " " " bold(w) = − 2 bold(i) + display(frac(7, 2)) bold(j)$].
]
]


      
        Find the coordinate form for the vector #math.equation(block: false, alt: "u equals 3 v")[$bold(u) = 3 bold(v)$], where #math.equation(block: false, alt: "v equals minus 12 i plus 15 j")[$bold(v) = − 12 bold(i) + 15 bold(j)$].


        
          #math.equation(block: true, alt: "u equals minus 36 i plus 45 j")[$bold(u) = − 36 bold(i) + 45 bold(j)$]


        
      
      If we divide a non-zero vector #math.equation(block: false, alt: "v")[$bold(v)$] by its own length, we create a unit vector that points in the same direction as #math.equation(block: false, alt: "v")[$bold(v)$].  (Dividing a vector by a scalar #math.equation(block: false, alt: "k")[$k$] is the same as multiplying the vector by #math.equation(block: false, alt: "the fraction 1 over k")[$display(frac(1, k))$].)


      #examplebox("Example 5")[][
Find a unit vector pointing in the same direction as #math.equation(block: false, alt: "v equals 2 i minus 4 j")[$bold(v) = 2 bold(i) − 4 bold(j)$].


        
        #solutionbox[
We first compute the magnitude of #math.equation(block: false, alt: "v")[$bold(v)$].


          #math.equation(block: true, alt: "∥ v ∥ equals the square root of 2 squared plus open parenthesis minus 4 close parenthesis squared equals the square root of 20 equals 2 the square root of 5")[$∥ bold(v) ∥ = sqrt(2^(2) + ( − 4 )^(2)) = sqrt(20) = 2 sqrt(5)$]
          Next we multiply #math.equation(block: false, alt: "v")[$bold(v)$] by the reciprocal of its length.  The unit vector in the same direction as #math.equation(block: false, alt: "v")[$bold(v)$] is then


          #math.equation(block: true, alt: "u equals the fraction 1 over ∥ v ∥ v, equals the fraction 1 over 2 the square root of 5 open parenthesis 2 i minus 4 j close parenthesis; equals the fraction 1 over 2 the square root of 5 times 2 i plus the fraction 1 over 2 the square root of 5 times open parenthesis minus 4 j close parenthesis equals the fraction 1 over the square root of 5 i plus the fraction minus 2 over the square root of 5 j")[$bold(u) = display(frac(1, ∥ bold(v) ∥)) bold(v) & = display(frac(1, 2 sqrt(5))) ( 2 bold(i) − 4 bold(j) ) \  & = display(frac(1, 2 sqrt(5))) ⋅ 2 bold(i) + display(frac(1, 2 sqrt(5))) ⋅ ( − 4 bold(j) ) = display(frac(1, sqrt(5))) bold(i) + display(frac(− 2, sqrt(5))) bold(j)$]
]
]


      By computing its length, you can check that the vector #math.equation(block: false, alt: "u")[$bold(u)$] found in the previous Example really is a unit vector.


      #math.equation(block: true, alt: "∥ u ∥ equals the square root of open parenthesis the fraction 1 over the square root of 5 close parenthesis squared plus open parenthesis the fraction minus 2 over the square root of 5 close parenthesis squared equals the square root of the fraction 1 over 5 plus the fraction 4 over 5 equals the square root of 1 equals 1")[$∥ bold(u) ∥ = sqrt(attach(( display(frac(1, sqrt(5))) ), t: 2) + attach(( display(frac(− 2, sqrt(5))) ), t: 2)) = sqrt(display(frac(1, 5)) + display(frac(4, 5))) = sqrt(1) = 1$]
      Once we have a unit vector #math.equation(block: false, alt: "u")[$bold(u)$] that points in a given direction, we can create a vector of any length in that direction, simply by scaling #math.equation(block: false, alt: "u")[$bold(u)$] by the length we desire. For example, the vector of length 10 pointing in the same direction as #math.equation(block: false, alt: "v")[$bold(v)$] in the previous example is


      #math.equation(block: true, alt: "10 u, equals 10 times open parenthesis the fraction 1 over the square root of 5 i plus the fraction minus 2 over the square root of 5 j close parenthesis; equals 10 times the fraction 1 over the square root of 5 i plus 10 times the fraction minus 2 over the square root of 5 j; equals 2 the square root of 5 i minus 4 the square root of 5 i")[$10 bold(u) & = 10 ⋅ ( display(frac(1, sqrt(5))) bold(i) + display(frac(− 2, sqrt(5))) bold(j) ) \  & = 10 ⋅ display(frac(1, sqrt(5))) bold(i) + 10 ⋅ display(frac(− 2, sqrt(5))) bold(j) \  & = 2 sqrt(5) bold(i) − 4 sqrt(5) bold(i)$]
      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Scaling a Vector]


        A unit vector #math.equation(block: false, alt: "u")[$bold(u)$] in the direction of #math.equation(block: false, alt: "v")[$bold(v)$] is given by #math.equation(block: false, alt: "u equals the fraction 1 over ∥ v ∥ v .")[$" " " " bold(u) = display(frac(1, ∥ bold(v) ∥)) bold(v) .$]


        A vector #math.equation(block: false, alt: "w")[$bold(w)$] of length #math.equation(block: false, alt: "k")[$k$] in the direction of #math.equation(block: false, alt: "v")[$bold(v)$] is given by #math.equation(block: false, alt: "w equals the fraction k over ∥ v ∥ v .")[$" " " " bold(w) = display(frac(k, ∥ bold(v) ∥)) bold(v) .$]
]


      
        Find a unit vector #math.equation(block: false, alt: "u")[$bold(u)$] and a vector #math.equation(block: false, alt: "w")[$bold(w)$] of length 10 pointing in the same direction as #math.equation(block: false, alt: "v equals minus i plus 3 j")[$bold(v) = − bold(i) + 3 bold(j)$].


        
          #math.equation(block: true, alt: "u equals minus the fraction 1 over the square root of 10 i plus the fraction 3 over the square root of 10 j")[$bold(u) = − display(frac(1, sqrt(10))) bold(i) + display(frac(3, sqrt(10))) bold(j)$], #math.equation(block: true, alt: "w equals minus the square root of 10 i plus 3 the square root of 10 j")[$" " bold(w) = − sqrt(10) bold(i) + 3 sqrt(10) bold(j)$]


        
      
    
    
      
=== Adding Vectors in Coordinate Form


      It is also easy to add vectors in coordinate form. The figure below shows the sum of #math.equation(block: false, alt: "u equals i plus 2 j")[$bold(u) = bold(i) + 2 bold(j)$] and  #math.equation(block: false, alt: "v equals 5 i plus 3 j")[$bold(v) = 5 bold(i) + 3 bold(j)$]. Remember that we add two vectors by following the first vector by the second.


      The vector sum describes the path


      #math.equation(block: true, alt: "1 unit in the x -direction, then; 2 units in the y -direction,; followed by; 5 units in the x -direction, then; 3 units in the y -direction.")[$1 " " "unit in the " x "-direction, then" \ 2 " " "units in the " y "-direction," \ "followed by" \ 5 " " "units in the " x "-direction, then" \ 3 " " "units in the " y "-direction. "$]
      But we arrive at the same endpoint by traveling


      #math.equation(block: true, alt: "1 plus 5 equals 6 units in the x -direction, then; 2 plus 3 equals 5 units in the y -direction.")[$1 + 5 = 6 " " "units in the " x "-direction, then" \ 2 + 3 = 5 " " "units in the " y "-direction."$]
      #figure(figph[triangle], alt: "triangle", caption: none)


      The resultant vector has components that are just the sums of the components of the vectors #math.equation(block: false, alt: "u")[$bold(u)$] and #math.equation(block: false, alt: "v")[$bold(v)$]. To add two vectors in coordinate form, we add the corresponding components.


      The rule above, drawn tip-to-tail. The blue vector is fixed at u = i + 2j; the red vector v = ci + dj follows it, with its components c and d on the sliders; and the bright arrow from the origin is the resultant u + v, its endpoint marked with a dot. At the starting values v = 5i + 3j, this is exactly the book's figure: travel 1 right and 2 up, then 5 right and 3 up, and you arrive at (6, 5) — the components add. Slide c and d anywhere and the dot always lands at (1 + c, 2 + d). One setting deserves a special look: at v = −i − 2j, the exact opposite of u, the resultant collapses to the zero vector and the dot sits at the origin — the u + v = 0 case the section discusses.{"functions":\[{"color":"\#1f4e79","expression":"t\*1","expression2":"t\*2","id":"vector-u","inequality":"lt","lineDash":\[\],"lineWidth":3,"mode":"parametric","tMax":1,"tMin":0,"variable":"t","visible":true},{"color":"\#b8442a","expression":"1 + t\*c","expression2":"2 + t\*d","id":"vector-v","inequality":"lt","lineDash":\[\],"lineWidth":3,"mode":"parametric","tMax":1,"tMin":0,"variable":"t","visible":true},{"color":"\#2563eb","expression":"t\*(1 + c)","expression2":"t\*(2 + d)","id":"resultant","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"parametric","tMax":1,"tMin":0,"variable":"t","visible":true},{"color":"\#2563eb","expression":"1 + c + 0.14\*cos(t)","expression2":"2 + d + 0.14\*sin(t)","id":"sum-dot","inequality":"lt","lineDash":\[\],"lineWidth":3,"mode":"parametric","tMax":6.2832,"tMin":0,"variable":"t","visible":true}\],"grid":{"majorSpacing":1,"showAxes":true,"showLabels":true},"parameters":\[{"id":"p-c","label":"Component c of v (east-west)","max":6,"min":-5,"name":"c","step":1,"value":5},{"id":"p-d","label":"Component d of v (north-south)","max":4,"min":-4,"name":"d","step":1,"value":3}\],"title":"u + v: u = i + 2j fixed, v = ci + dj on sliders","version":1,"viewport":{"centerX":1.5,"centerY":1.5,"scale":40}}#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Sum of Vectors in Coordinate Form]


        If #math.equation(block: false, alt: "u equals a i plus b j")[$bold(u) = a bold(i) + b bold(j)$] and #math.equation(block: false, alt: "v equals c i plus d j")[$bold(v) = c bold(i) + d bold(j)$], then


        #math.equation(block: true, alt: "u plus v equals open parenthesis a plus c close parenthesis i plus open parenthesis b plus d close parenthesis j")[$bold(u) + bold(v) = ( bold(italic(a)) + bold(italic(c)) ) bold(i) + ( bold(italic(b)) + bold(italic(d)) ) bold(j)$]
]


      #examplebox("Example 6")[][
Kyle flew 15 miles southwest and then 20 miles on a bearing of #math.equation(block: false, alt: "260 degrees")[$260^(∘)$]. What is his current position relative to his starting point?


        
        #solutionbox[
A sketch of Kyle's journey is shown below. We begin by converting the vector for each leg of his journey into coordinate form.


          #math.equation(block: true, alt: "v sub x, equals 15 cos ⁡ open parenthesis 225 degrees close parenthesis equals minus 10.607; v sub y, equals 15 sin ⁡ open parenthesis 225 degrees close parenthesis equals minus 10.607; w sub x, equals 20 cos ⁡ open parenthesis 190 degrees close parenthesis equals minus 19.696; w sub y, equals 20 sin ⁡ open parenthesis 190 degrees close parenthesis equals minus 3.473")[$v_(x) & = 15 cos ( 225^(∘) ) = − 10.607 \ v_(y) & = 15 sin ( 225^(∘) ) = − 10.607 \ w_(x) & = 20 cos ( 190^(∘) ) = − 19.696 \ w_(y) & = 20 sin ( 190^(∘) ) = − 3.473$]
          #figure(figph[vectors], alt: "vectors", caption: none)


          Thus,


          #math.equation(block: true, alt: "v equals minus 10.607 i minus 10.607 j and w equals minus 19.696 i minus 3.473 j")[$bold(v) = − 10.607 bold(i) − 10.607 bold(j) " " " " " " " " "and" " " " " " " " " bold(w) = − 19.696 bold(i) − 3.473 bold(j)$]
          We find the resultant vector, #math.equation(block: false, alt: "r")[$bold(r)$], by adding #math.equation(block: false, alt: "v")[$bold(v)$] and #math.equation(block: false, alt: "w")[$bold(w)$].


          #math.equation(block: true, alt: "r, equals open parenthesis minus 10.607 minus 19.696 close parenthesis i plus open parenthesis minus 10.607 minus 3.473 close parenthesis j; equals minus 30.303 i minus 14.08 j")[$bold(r) & = ( − 10.607 − 19.696 ) bold(i) + ( − 10.607 − 3.473 ) bold(j) \  & = − 30.303 bold(i) − 14.08 bold(j)$]
          Kyle's position vector is #math.equation(block: false, alt: "r equals 30.303 i minus 14.08 j")[$bold(r) = 30.303 bold(i) − 14.08 bold(j)$], or about 30.3 miles west and 14.1 miles south of his starting position.
]
]


      
        Arianna's yacht heads #math.equation(block: false, alt: "10 degrees")[$10^(∘)$] west of north at a speed of 20 kilometers per hour relative to the water. However, the water is moving #math.equation(block: false, alt: "70 degrees")[$70^(∘)$] east of north at 4 kilometers per hour. What is Arianna's velocity relative to land?


        
          21.1 kph, #math.equation(block: false, alt: "0.8 degrees")[$0.8^(∘)$] east of due north


        
      
      The coordinate form is especially efficient if we want to add more than two vectors.


      #examplebox("Example 7")[][
While exploring an abandoned cottage in the woods, you discover an old map showing the location of a buried treasure.


          + From the cottage, go 10 km in the direction #math.equation(block: false, alt: "70 degrees")[$70^(∘)$].
+ From that point, go 12.8 km in the direction #math.equation(block: false, alt: "348 degrees")[$348^(∘)$].
+ Finally, go 9.5 km in the direction #math.equation(block: false, alt: "129 degrees")[$129^(∘)$].


          You wonder if the directions will lead you safely to the treasure. You draw vectors to illustrate the path described by the directions, as shown in the figure.


          #figure(figph[vectors on map], alt: "vectors on map", caption: none)


          Instead of following the map directions, you plan to fly by helicopter from the abandoned cottage directly to the treasure, but you will need to know the correct heading and distance to fly. Draw the vector for the net displacement from the cottage to the treasure, and give its length and direction.


        
        #solutionbox[
The helicopter's flight plan is the vector sum of the three vectors #math.equation(block: false, alt: "u")[$bold(u)$], #math.equation(block: false, alt: "v")[$bold(v)$], and #math.equation(block: false, alt: "w")[$bold(w)$] shown in the figure. To find their sum, we resolve each of the three vectors into its components. In coordinate form, the three displacement vectors are


          #math.equation(block: true, alt: "u, equals open parenthesis 10 cos ⁡ open parenthesis 70 degrees close parenthesis close parenthesis i plus open parenthesis 10 sin ⁡ open parenthesis 70 degrees close parenthesis close parenthesis j equals 3.42 i plus 9.40 j; v, equals open parenthesis 12.8 cos ⁡ open parenthesis 348 degrees close parenthesis close parenthesis i plus open parenthesis 12.8 sin ⁡ open parenthesis 348 degrees close parenthesis close parenthesis j equals 12.52 i minus 2.66 j; w, equals open parenthesis 9.5 cos ⁡ open parenthesis 129 degrees close parenthesis close parenthesis i plus open parenthesis 9.5 sin ⁡ open parenthesis 129 degrees close parenthesis close parenthesis j equals minus 5.98 i plus 7.38 j")[$bold(u) & = ( 10 cos ( 70^(∘) ) ) bold(i) + ( 10 sin ( 70^(∘) ) ) bold(j) = 3.42 bold(i) + 9.40 bold(j) \ bold(v) & = ( 12.8 cos ( 348^(∘) ) ) bold(i) + ( 12.8 sin ( 348^(∘) ) ) bold(j) = 12.52 bold(i) − 2.66 bold(j) \ bold(w) & = ( 9.5 cos ( 129^(∘) ) ) bold(i) + ( 9.5 sin ( 129^(∘) ) ) bold(j) = − 5.98 bold(i) + 7.38 bold(j)$]
          The net displacement is given by the resultant vector, #math.equation(block: false, alt: "r")[$bold(r)$]. We add the corresponding components of#math.equation(block: false, alt: "u")[$bold(u)$], #math.equation(block: false, alt: "v")[$bold(v)$], and #math.equation(block: false, alt: "w")[$bold(w)$].


          #math.equation(block: true, alt: "r, equals open parenthesis 3.42 i plus 9.40 j close parenthesis plus open parenthesis 12.52 i minus 2.66 j close parenthesis plus open parenthesis minus 5.98 i plus 7.38 j close parenthesis; equals open parenthesis 3.42 plus 12.52 minus 5.98 close parenthesis i plus open parenthesis 9.40 minus 2.66 plus 7.38 close parenthesis j equals 10 i plus 14.12 j")[$bold(r) & = ( 3.42 bold(i) + 9.40 bold(j) ) + ( 12.52 bold(i) − 2.66 bold(j) ) + ( − 5.98 bold(i) + 7.38 bold(j) ) \  & = ( 3.42 + 12.52 − 5.98 ) bold(i) + ( 9.40 − 2.66 + 7.38 ) bold(j) = 10 bold(i) + 14.12 bold(j)$]
          The treasure is at the point 10 km east and 14.12 km north of the cottage. To find the flight plan for the helicopter, we compute the magnitude and direction of the vector #math.equation(block: false, alt: "r")[$bold(r)$].


          #math.equation(block: true, alt: "∥ r ∥, equals the square root of 10 squared plus 14.12 squared equals 17.3; tan ⁡ open parenthesis θ close parenthesis, equals the fraction 14.12 over 10 equals 1.412; θ, equals tan to the power minus 1 ⁡ open parenthesis 1.412 close parenthesis equals 54.69 degrees")[$∥ bold(r) ∥ & = sqrt(10^(2) + 14.12^(2)) = 17.3 \ tan ( θ ) & = display(frac(14.12, 10)) = 1.412 \ θ & = tan^(− 1) ( 1.412 ) = 54.69^(∘)$]
          Thus, the treasure is 17.3 km from the cottage, in the direction #math.equation(block: false, alt: "54.7 degrees")[$54.7^(∘)$] from east. The helicopter's flight path is shown at right.


          #figure(figph[one vector on map], alt: "one vector on map", caption: none)
]
]


      
        From your campsite you hike 1.6 km in the direction #math.equation(block: false, alt: "175 degrees")[$175^(∘)$], then 0.8 km in the direction #math.equation(block: false, alt: "65 degrees")[$65^(∘)$], and finally 1.2 km in the direction #math.equation(block: false, alt: "350 degrees")[$350^(∘)$].


        + Draw a diagram for your hike, using vectors to represent each of the three segments.
+ Resolve each vector into components, and determine your location after your hike.


        
          + #figure(figph[vectors], alt: "vectors", caption: none)
+ The three vectors describing the hike are #math.equation(block: false, alt: "0.139 i minus 1.594 j")[$0.139 bold(i) − 1.594 bold(j)$], #math.equation(block: false, alt: "0.725 i plus 0.338 j")[$" " 0.725 bold(i) + 0.338 bold(j)$], and #math.equation(block: false, alt: "minus 0.208 i plus 1.182 j")[$" " − 0.208 bold(i) + 1.182 bold(j)$]. #linebreak() Your final locations is #math.equation(block: false, alt: "0.656 i minus 0.074 j")[$" " 0.656 bold(i) − 0.074 bold(j)$].


        
      
    
    
      
=== Force


      When you push or pull on something, you are exerting a #strong[force] on the object. For example, the weight of an object is actually a force, the result of gravity pulling the object towards the earth. A force has magnitude (measured in pounds) and direction, so force is a vector quantity. A force applied to an object causes the object to accelerate in the direction of the force.


      When two or more forces act simultaneously on an object, the object moves as if it were acted on by the sum of the individual force vectors. The sum of all the forces acting on an object is called the #strong[resultant force].


      #examplebox("Example 8")[][
Abe and Bart are trying to move a refrigerator by pulling on it with forces of 150 pounds and 120 pounds respectively, as shown in figure (a).  What is the resulting force on the refrigerator, and in what direction will it move?


          #figure(figph[vectors], alt: "vectors", caption: none)


        
        #solutionbox[
We write each force in its coordinate form.


          #math.equation(block: true, alt: "A, equals open parenthesis 150 cos ⁡ open parenthesis 20 degrees close parenthesis close parenthesis i plus open parenthesis 150 sin ⁡ open parenthesis 20 degrees close parenthesis close parenthesis j equals 140.95 i plus 51.30 j; B, equals open parenthesis 120 cos ⁡ open parenthesis 80 degrees close parenthesis close parenthesis i plus open parenthesis 120 sin ⁡ open parenthesis 80 degrees close parenthesis close parenthesis j equals 20.84 i plus 118.18 j")[$bold(A) & = ( 150 cos ( 20^(∘) ) ) bold(i) + ( 150 sin ( 20^(∘) ) ) bold(j) = 140.95 bold(i) + 51.30 bold(j) \ bold(B) & = ( 120 cos ( 80^(∘) ) ) bold(i) + ( 120 sin ( 80^(∘) ) ) bold(j) = 20.84 bold(i) + 118.18 bold(j)$]
          Next, we add the forces applied by Abe and Bart to find the resultant force, #math.equation(block: false, alt: "r equals A plus B")[$bold(r) = bold(A) + bold(B)$].


          #math.equation(block: true, alt: "r, equals open parenthesis 140.95 i plus 51.30 j close parenthesis plus open parenthesis 20.84 i plus 118.18 j close parenthesis; equals 161.75 i plus 169.48 j")[$bold(r) & = ( 140.95 bold(i) + 51.30 bold(j) ) + ( 20.84 bold(i) + 118.18 bold(j) ) \  & = 161.75 bold(i) + 169.48 bold(j)$]
          The magnitude of the resultant force is


          #math.equation(block: true, alt: "∥ r ∥ equals the square root of 161.75 squared plus 169.48 squared equals 234.28")[$∥ bold(r) ∥ = sqrt(161.75^(2) + 169.48^(2)) = 234.28$]
          and the direction of the force is given by


          #math.equation(block: true, alt: "tan to the power minus 1 ⁡ open parenthesis the fraction 169.48 over 161.75 close parenthesis equals 46.33 degrees")[$tan^(− 1) ( display(frac(169.48, 161.75)) ) = 46.33^(∘)$]
          #figure(figph[vectors], alt: "vectors", caption: none)


          The force exerted on the refrigerator is about 234 pounds, and it will move at an angle of #math.equation(block: false, alt: "46 degrees")[$46^(∘)$] from the horizontal segment shown in the figure.
]
]


      
        Two tugboats pull on a barge in the river with forces #math.equation(block: false, alt: "u equals 20 i plus 8 j")[$bold(u) = 20 bold(i) + 8 bold(j)$] and #math.equation(block: false, alt: "v equals 28 i minus 6 j")[$bold(v) = 28 bold(i) − 6 bold(j)$], measured in thousands of pounds. In what direction will the barge move, and what is the magnitude of the force propelling it?


        
          #math.equation(block: false, alt: "2.4 degrees ,")[$2.4^(∘) , " "$]48.04 thousand lbs


        
      
      If vectors #math.equation(block: false, alt: "u")[$bold(u)$] and #math.equation(block: false, alt: "v")[$bold(v)$] have the same magnitude but opposite directions, then #math.equation(block: false, alt: "u plus v")[$bold(u) + bold(v)$] has zero magnitude and is called the #strong[zero vector]. A zero displacement vector means that the object started and ended in the same place; a zero velocity vector means that the object is not moving. We denote the zero vector by #math.equation(block: false, alt: "0")[$0$], so #math.equation(block: false, alt: "∥ 0 ∥ equals 0")[$∥ 0 ∥ = 0$].


      The vector that has the same magnitude as #math.equation(block: false, alt: "v")[$bold(v)$] but the opposite direction is called the #strong[opposite] of #math.equation(block: false, alt: "v")[$bold(v)$] and denoted by #math.equation(block: false, alt: "minus v")[$− bold(v)$].  Then


      #math.equation(block: true, alt: "v plus minus v equals 0")[$bold(v) + − bold(v) = 0$]
      If the sum of the forces acting on an object is #math.equation(block: false, alt: "0")[$0$], the forces are said to be in #strong[equilibrium], and the object will remain stationary.


      #examplebox("Example 9")[][
Recall the refrigerator in the previous example. Carl does not want Abe and Bart to move the refrigerator. How hard must Carl pull so that the refrigerator remains motionless?


        
        #solutionbox[
The force #math.equation(block: false, alt: "C")[$bold(C)$] that Carl applies must equal in magnitude the sum of the forces applied by Abe and Bart, but point in the opposite direction.  That is,


          #math.equation(block: true, alt: "C equals minus r equals minus open parenthesis A plus B close parenthesis")[$bold(C) = − bold(r) = − ( bold(A) + bold(B) )$]
          So Carl must pull with 234.3 pounds in the direction #math.equation(block: false, alt: "180 degrees plus 46.33 degrees equals 226.33 degrees")[$180^(∘) + 46.33^(∘) = 226.33^(∘)$].
]
]


      
        Suppose that #math.equation(block: false, alt: "u plus v plus w equals 0 ,")[$" " " " bold(u) + bold(v) + bold(w) = 0 , " " " "$] and that #math.equation(block: false, alt: "u equals minus 8 i plus 32 j")[$bold(u) = − 8 bold(i) + 32 bold(j)$] and #math.equation(block: false, alt: "v equals 14 i minus 26 j .")[$bold(v) = 14 bold(i) − 26 bold(j) . " " " "$] What is the coordinate form of vector #math.equation(block: false, alt: "w")[$bold(w)$]?


        
          #math.equation(block: true, alt: "w equals minus 6 i minus 6 j")[$bold(w) = − 6 bold(i) − 6 bold(j)$]


        
      
      Review the following skills you will need for this section.


      #notebox("Algebra Refresher", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
+ State the sum of angles formula for sine.
+ State the sum of angles formula for cosine.
+ State the sum of angles formula for tangent.
+ Explain how the difference of angles formulas differ from the sum of angles formulas. #linebreak() For Problems 5 and 6, let #math.equation(block: false, alt: "cos ⁡ open parenthesis α close parenthesis equals the fraction minus 2 over the square root of 5 , 90 degrees less than α less than 180 degrees")[$cos ( α ) = display(frac(− 2, sqrt(5))) , " " 90^(∘) < α < 180^(∘)$].
+ Find an exact value for #math.equation(block: false, alt: "cos ⁡ open parenthesis α plus 30 degrees close parenthesis")[$cos ( α + 30^(∘) )$].
+ Find an exact value for #math.equation(block: false, alt: "tan ⁡ open parenthesis α minus 30 degrees close parenthesis")[$tan ( α − 30^(∘) )$].


        #math.equation(block: true, alt: "bar")[$underline(#h(2em) #h(2em) #h(2em) #h(2em))$]


        Skills Refresher Answers


        + #math.equation(block: false, alt: "sin ⁡ open parenthesis α plus β close parenthesis equals sin ⁡ open parenthesis α close parenthesis cos ⁡ open parenthesis β close parenthesis plus cos ⁡ open parenthesis α close parenthesis sin ⁡ open parenthesis β close parenthesis")[$sin ( α + β ) = sin ( α ) cos ( β ) + cos ( α ) sin ( β )$]
+ #math.equation(block: false, alt: "cos ⁡ open parenthesis α plus β close parenthesis equals cos ⁡ open parenthesis α close parenthesis cos ⁡ open parenthesis β close parenthesis minus sin ⁡ open parenthesis α close parenthesis sin ⁡ open parenthesis β close parenthesis")[$cos ( α + β ) = cos ( α ) cos ( β ) − sin ( α ) sin ( β )$]
+ #math.equation(block: false, alt: "tan ⁡ open parenthesis α plus β close parenthesis equals the fraction tan ⁡ open parenthesis α close parenthesis plus tan ⁡ open parenthesis β close parenthesis over 1 tan ⁡ open parenthesis α close parenthesis tan ⁡ open parenthesis β close parenthesis")[$tan ( α + β ) = display(frac(tan ( α ) + tan ( β ), 1 tan ( α ) tan ( β )))$]
+ Change subtraction signs to addition and vice versa.
+ #math.equation(block: false, alt: "the fraction minus 2 the square root of 3 minus 1 over 2 the square root of 5")[$display(frac(− 2 sqrt(3) − 1, 2 sqrt(5)))$]
+ #math.equation(block: false, alt: "the fraction minus the square root of 3 minus 2 over 1 plus 2 the square root of 3")[$display(frac(− sqrt(3) − 2, 1 + 2 sqrt(3)))$]
]


    
    
      
=== Section 9.2 Summary


      
        
==== Vocabulary


        - Unit vector
- Coordinate form
- Geometric form
- Zero vector


      
      
        
==== Concepts


        + A vector of magnitude 1 is called a #strong[unit vector]. The unit vector in the direction of the #math.equation(block: false, alt: "x")[$x$]-axis is denoted by #math.equation(block: false, alt: "i")[$bold(i)$]. The unit vector in the direction of the #math.equation(block: false, alt: "y")[$y$]-axis is called #math.equation(block: false, alt: "j")[$bold(j)$].
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Coordinate Form of a Vector]

The vector

#math.equation(block: true, alt: "v equals a i plus b j")[$bold(v) = a bold(i) + b bold(j)$]is the vector whose horizontal component is #math.equation(block: false, alt: "a")[$a$] and whose vertical component is #math.equation(block: false, alt: "b")[$b$].
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Comparing the Geometric and Coordinate Forms of a Vector]

Suppose that the vector #math.equation(block: false, alt: "v")[$bold(v)$] has magnitude #math.equation(block: false, alt: "∥ v ∥")[$∥ bold(v) ∥$] and points in the direction of the angle #math.equation(block: false, alt: "θ")[$θ$] in standard position. If #math.equation(block: false, alt: "v")[$bold(v)$] has the coordinate form #math.equation(block: false, alt: "v equals a i plus b j")[$bold(v) = a bold(i) + b bold(j)$], then

#math.equation(block: true, alt: "a, equals ∥ v ∥ cos ⁡ θ, ∥ v ∥, equals the square root of a squared plus b squared; b, equals ∥ v ∥ sin ⁡ θ, tan ⁡ θ, equals the fraction b over a")[$a & = ∥ bold(v) ∥ cos θ &  &  & ∥ bold(v) ∥ & = sqrt(a^(2) + b^(2)) \ b & = ∥ bold(v) ∥ sin θ &  &  & tan θ & = display(frac(b, a))$]
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Scalar Multiplication in Coordinate Form]

If #math.equation(block: false, alt: "v equals a i plus b j")[$bold(v) = a bold(i) + b bold(j)$] and #math.equation(block: false, alt: "k")[$k$] is a scalar, then

#math.equation(block: true, alt: "k v equals k a i plus k b j")[$k bold(v) = k a bold(i) + k b bold(j)$]
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Sum of Vectors in Coordinate Form]

If #math.equation(block: false, alt: "u equals a i plus b j")[$bold(u) = a bold(i) + b bold(j)$] and #math.equation(block: false, alt: "v equals c i plus d j")[$bold(v) = c bold(i) + d bold(j)$], then

#math.equation(block: true, alt: "u plus v equals open parenthesis a plus c close parenthesis i plus open parenthesis b plus d close parenthesis j")[$bold(u) + bold(v) = ( a + c ) bold(i) + ( b + d ) bold(j)$]
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Scaling a Vector]

A unit vector #math.equation(block: false, alt: "u")[$bold(u)$] in the direction of #math.equation(block: false, alt: "v")[$bold(v)$] is given by #math.equation(block: false, alt: "u equals the fraction 1 over ∥ v ∥ v .")[$bold(u) = display(frac(1, ∥ bold(v) ∥)) bold(v) .$]

A vector #math.equation(block: false, alt: "w")[$bold(w)$] of length #math.equation(block: false, alt: "k")[$k$] in the direction of #math.equation(block: false, alt: "v")[$bold(v)$] is given by #math.equation(block: false, alt: "w equals the fraction k over ∥ v ∥ v .")[$bold(w) = display(frac(k, ∥ bold(v) ∥)) bold(v) .$]
]


      
      
        
====  Study Questions


        + Is it easier to add vectors in geometric form or in coordinate form? Why?
+ To find a unit vector in the direction of #math.equation(block: false, alt: "v")[$bold(v)$], multiply the coordinates of #math.equation(block: false, alt: "v")[$bold(v)$] 
      by \_\_\_\_\_\_.
+ To find a vector of length #math.equation(block: false, alt: "k")[$k$] in the direction of #math.equation(block: false, alt: "v")[$bold(v)$], multiply the coordinates of #math.equation(block: false, alt: "v")[$bold(v)$] 
      by  \_\_\_\_\_\_.
+ Name two physical quantities that are represented by vectors.


      
      
        
==== Skills


        + Convert the coordinate form of a vector to geometric form    \#7–18
+ Convert the geometric form of a vector to coordinate form    \#19–22, 47–50
+ Compute sums and scalar multiples of vectors    \#1–8, 23–28, 47–50
+ Find a vector in a given direction with a given length    \#39–46
+ Solve problems with vectors    \#51–60


      
    
    
      
=== Homework 9-2


      For Problems 1–4, give the coordinate form of each vector shown in the figure. Use the coordinate form to find the following.


      #figure(figph[vectors], alt: "vectors", caption: none)


      
        + #math.equation(block: false, alt: "∥ u ∥")[$∥ bold(u) ∥$]
+ #math.equation(block: false, alt: "2 u")[$2 bold(u)$]
+ #math.equation(block: false, alt: "∥ 2 u ∥")[$∥ 2 bold(u) ∥$]


        
          #math.equation(block: true, alt: "u equals 3 i plus 2 j")[$bold(u) = 3 bold(i) + 2 bold(j)$]


          + #math.equation(block: false, alt: "the square root of 13")[$sqrt(13)$]
+ #math.equation(block: false, alt: "6 i plus 4 j")[$6 bold(i) + 4 bold(j)$]
+ #math.equation(block: false, alt: "2 the square root of 13")[$2 sqrt(13)$]


        
      
      
        + #math.equation(block: false, alt: "∥ v ∥")[$∥ bold(v) ∥$]
+ #math.equation(block: false, alt: "the fraction 1 over 2 v")[$display(frac(1, 2)) bold(v)$]
+ #math.equation(block: false, alt: "‖ the fraction 1 over 2 v ‖")[$‖ display(frac(1, 2)) bold(v) ‖$]


      
      
        + #math.equation(block: false, alt: "∥ w ∥")[$∥ bold(w) ∥$]
+ #math.equation(block: false, alt: "minus w")[$− bold(w)$]
+ #math.equation(block: false, alt: "∥ minus w ∥")[$∥ − bold(w) ∥$]


        
          #math.equation(block: true, alt: "w equals 6 i minus 3 j")[$bold(w) = 6 bold(i) − 3 bold(j)$]


          + #math.equation(block: false, alt: "3 the square root of 5")[$3 sqrt(5)$]
+ #math.equation(block: false, alt: "minus 6 i plus 3 j")[$− 6 bold(i) + 3 bold(j)$]
+ #math.equation(block: false, alt: "3 the square root of 5")[$3 sqrt(5)$]


        
      
      
        + #math.equation(block: false, alt: "∥ z ∥")[$∥ bold(z) ∥$]
+ #math.equation(block: false, alt: "minus 3 z")[$− 3 bold(z)$]
+ #math.equation(block: false, alt: "∥ minus 3 z ∥")[$∥ − 3 bold(z) ∥$]


      
      
        + For the vectors #math.equation(block: false, alt: "u")[$bold(u)$] and #math.equation(block: false, alt: "v")[$bold(v)$] above, calculate #math.equation(block: false, alt: "u plus v")[$bold(u) + bold(v)$] and #math.equation(block: false, alt: "∥ u plus v ∥")[$∥ bold(u) + bold(v) ∥$].
+ Which of the following statements is true?+ #math.equation(block: false, alt: "∥ u ∥ plus ∥ v ∥ less than or equal to ∥ u plus v ∥")[$∥ bold(u) ∥ + ∥ bold(v) ∥ ≤ ∥ bold(u) + bold(v) ∥$]
+ #math.equation(block: false, alt: "∥ u ∥ plus ∥ v ∥ equals ∥ u plus v ∥")[$∥ bold(u) ∥ + ∥ bold(v) ∥ = ∥ bold(u) + bold(v) ∥$]
+ #math.equation(block: false, alt: "∥ u ∥ plus ∥ v ∥ greater than or equal to ∥ u plus v ∥")[$∥ bold(u) ∥ + ∥ bold(v) ∥ ≥ ∥ bold(u) + bold(v) ∥$]


        
          + #math.equation(block: false, alt: "u plus v equals minus 2 i plus 5 j")[$bold(u) + bold(v) = − 2 bold(i) + 5 bold(j)$] and #math.equation(block: false, alt: "∥ u plus v ∥ equals the square root of 29")[$∥ bold(u) + bold(v) ∥ = sqrt(29)$]
+ #math.equation(block: false, alt: "∥ u ∥ plus ∥ v ∥ greater than or equal to ∥ u plus v ∥")[$∥ bold(u) ∥ + ∥ bold(v) ∥ ≥ ∥ bold(u) + bold(v) ∥$]


        
      
      
        + For the vectors #math.equation(block: false, alt: "w")[$bold(w)$] and #math.equation(block: false, alt: "z")[$bold(z)$] above, calculate #math.equation(block: false, alt: "w plus z")[$bold(w) + bold(z)$] and #math.equation(block: false, alt: "∥ w plus z ∥")[$∥ bold(w) + bold(z) ∥$].
+ Which of the following statements is true?+ #math.equation(block: false, alt: "∥ w ∥ plus ∥ z ∥ less than or equal to ∥ w plus z ∥")[$∥ bold(w) ∥ + ∥ bold(z) ∥ ≤ ∥ bold(w) + bold(z) ∥$]
+ #math.equation(block: false, alt: "∥ w ∥ plus ∥ z ∥ equals ∥ w plus z ∥")[$∥ bold(w) ∥ + ∥ bold(z) ∥ = ∥ bold(w) + bold(z) ∥$]
+ #math.equation(block: false, alt: "∥ w ∥ plus ∥ z ∥ greater than or equal to ∥ w plus z ∥")[$∥ bold(w) ∥ + ∥ bold(z) ∥ ≥ ∥ bold(w) + bold(z) ∥$]


      
      For Problems 7–10,


      + Sketch the vector and give its coordinate form.
+ Find the magnitude and direction of the vector.


      #figure(figph[grid], alt: "grid", caption: none)


      
        The displacement vector from #math.equation(block: false, alt: "open parenthesis 1 , minus 2 close parenthesis")[$( 1 , − 2 )$] to #math.equation(block: false, alt: "open parenthesis minus 4 , 6 close parenthesis")[$( − 4 , 6 )$].


        
          + #figure(figph[vector], alt: "vector", caption: none)

#math.equation(block: true, alt: "minus 5 i plus 8 j")[$− 5 bold(i) + 8 bold(j)$]
+ #math.equation(block: false, alt: "∥ v ∥ equals the square root of 89 , θ equals 122 degrees")[$∥ bold(v) ∥ = sqrt(89) , " " " " θ = 122^(∘)$]


        
      
      
        The displacement vector from #math.equation(block: false, alt: "open parenthesis minus 5 , 2 close parenthesis")[$( − 5 , 2 )$] to #math.equation(block: false, alt: "open parenthesis 4 , 7 close parenthesis")[$( 4 , 7 )$].


      
      
        The displacement vector from #math.equation(block: false, alt: "open parenthesis minus 2 , 9 close parenthesis")[$( − 2 , 9 )$] to #math.equation(block: false, alt: "open parenthesis minus 4 , 8 close parenthesis")[$( − 4 , 8 )$].


        
          + #figure(figph[vector], alt: "vector", caption: none)

#math.equation(block: true, alt: "minus 2 i minus j")[$− 2 bold(i) − bold(j)$]
+ #math.equation(block: false, alt: "∥ v ∥ equals the square root of 5 , θ equals 206.6 degrees")[$∥ bold(v) ∥ = sqrt(5) , " " " " θ = 206.6^(∘)$]


        
      
      
        The displacement vector from #math.equation(block: false, alt: "open parenthesis minus 6 , 2 close parenthesis")[$( − 6 , 2 )$] to #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$].


      
      
        Hermione is 12 meters east and 3 meters north of Harry. Ron is 6 meters east and 9 meters north of Hermione.


        + Calculate the displacement vector from Harry to Ron in coordinate form. Let #math.equation(block: false, alt: "i")[$bold(i)$] point east and #math.equation(block: false, alt: "j")[$bold(j)$] point north.
+ Find the magnitude and direction of the displacement vector.


        
          + #math.equation(block: false, alt: "18 i plus 12 j")[$18 bold(i) + 12 bold(j)$]
+ #math.equation(block: false, alt: "∥ v ∥ equals 6 the square root of 13 , θ equals 33.7 degrees")[$∥ bold(v) ∥ = 6 sqrt(13) , " " " " θ = 33.7^(∘)$]


        
      
      
        Delbert and Francine are climbing a rock wall. Delbert is 8 feet to the right and and 23 feet above their starting point. Francine is 2 feet to the right and 7 feet above Delbert.


        + Calculate the displacement vector from the starting point to Francine in coordinate form. Let #math.equation(block: false, alt: "i")[$bold(i)$] point right and #math.equation(block: false, alt: "j")[$bold(j)$] point up.
+ Find the magnitude and direction of the displacement vector.


      
      For Problems 13–18, find the magnitude and direction of the vector.


      
        #math.equation(block: true, alt: "v equals minus 6 i plus 6 j")[$bold(v) = − 6 bold(i) + 6 bold(j)$]


        
          #math.equation(block: true, alt: "∥ v ∥ equals 6 the square root of 2 , θ equals 135 degrees")[$∥ bold(v) ∥ = 6 sqrt(2) , " " " " θ = 135^(∘)$]


        
      
      
        #math.equation(block: true, alt: "p equals minus 12 i minus 5 j")[$bold(p) = − 12 bold(i) − 5 bold(j)$]


      
      
        #math.equation(block: true, alt: "w equals 7 the square root of 3 i minus 7 j")[$bold(w) = 7 sqrt(3) bold(i) − 7 bold(j)$]


        
          #math.equation(block: true, alt: "∥ w ∥ equals 14 , θ equals minus 30 degrees")[$∥ bold(w) ∥ = 14 , " " " " θ = − 30^(∘)$]


        
      
      
        #math.equation(block: true, alt: "z equals minus 6 the square root of 2 i plus 6 the square root of 6 j")[$bold(z) = − 6 sqrt(2) bold(i) + 6 sqrt(6) bold(j)$]


      
      
        #math.equation(block: true, alt: "q equals 52 i plus 96 j")[$bold(q) = 52 bold(i) + 96 bold(j)$]


        
          #math.equation(block: true, alt: "∥ q ∥ equals 4 the square root of 745 , θ equals 61.56 degrees")[$∥ bold(q) ∥ = 4 sqrt(745) , " " " " θ = 61.56^(∘)$]


        
      
      
        #math.equation(block: true, alt: "s equals 3.2 i minus 1.8 j")[$bold(s) = 3.2 bold(i) − 1.8 bold(j)$]


      
      For Problems 19–22, find the coordinate form of the vector.


      
        #math.equation(block: true, alt: "∥ v ∥ equals 6 , θ equals minus 45 degrees")[$∥ bold(v) ∥ = 6 , " " θ = − 45^(∘)$]


        
          #math.equation(block: true, alt: "v equals 3 the square root of 2 i minus 3 the square root of 2 j")[$bold(v) = 3 sqrt(2) bold(i) − 3 sqrt(2) bold(j)$]


        
      
      
        #math.equation(block: true, alt: "∥ v ∥ equals 200 , θ equals 240 degrees")[$∥ bold(v) ∥ = 200 , " " θ = 240^(∘)$]


      
      
        #math.equation(block: true, alt: "∥ v ∥ equals 8.3 , θ equals 37 degrees")[$∥ bold(v) ∥ = 8.3 , " " θ = 37^(∘)$]


        
          #math.equation(block: true, alt: "v approximately equals 6.629 i plus 4.995 j")[$bold(v) ≈ 6.629 bold(i) + 4.995 bold(j)$]


        
      
      
        #math.equation(block: true, alt: "∥ v ∥ equals 23 , θ equals 200 degrees")[$∥ bold(v) ∥ = 23 , " " θ = 200^(∘)$]


      
      For Problems 23–26, sketch each vector and its components. Use the coordinate form to find the resultant vector #math.equation(block: false, alt: "u plus v")[$bold(u) + bold(v)$], and sketch it.


      
        #math.equation(block: true, alt: "u equals minus 3 i plus 2 j , v equals 4 i minus 4 j")[$bold(u) = − 3 bold(i) + 2 bold(j) , " " bold(v) = 4 bold(i) − 4 bold(j)$]


        
          #math.equation(block: true, alt: "i minus 2 j")[$bold(i) − 2 bold(j)$]


          #figure(figph[vectors], alt: "vectors", caption: none)


        
      
      
        #math.equation(block: true, alt: "u equals 5 i plus j , v equals 2 i minus 3 j")[$bold(u) = 5 bold(i) + bold(j) , " " bold(v) = 2 bold(i) − 3 bold(j)$]


      
      
        #math.equation(block: true, alt: "u equals minus 5 i minus 2 j , v equals i plus 6 j")[$bold(u) = − 5 bold(i) − 2 bold(j) , " " bold(v) = bold(i) + 6 bold(j)$]


        
          #math.equation(block: true, alt: "minus 4 i plus 4 j")[$− 4 bold(i) + 4 bold(j)$]


          #figure(figph[vectors], alt: "vectors", caption: none)


        
      
      
        #math.equation(block: true, alt: "u equals 8 i minus 3 j , v equals minus 4 i minus 2 j")[$bold(u) = 8 bold(i) − 3 bold(j) , " " bold(v) = − 4 bold(i) − 2 bold(j)$]


      
      For Problems 27–30, find the sum #math.equation(block: false, alt: "u plus v")[$bold(u) + bold(v)$] of the given vectors.


      
        #math.equation(block: true, alt: "u equals 13 i minus 8 j , v equals minus 1 i plus 11 j")[$bold(u) = 13 bold(i) − 8 bold(j) , " " bold(v) = − 1 bold(i) + 11 bold(j)$]


        
          #math.equation(block: true, alt: "12 i plus 3 j")[$12 bold(i) + 3 bold(j)$]


        
      
      
        #math.equation(block: true, alt: "u equals 3.7 i plus 2.6 j , v equals minus 1.3 i minus 5.7 j")[$bold(u) = 3.7 bold(i) + 2.6 bold(j) , " " bold(v) = − 1.3 bold(i) − 5.7 bold(j)$]


      
      
        #math.equation(block: true, alt: "u equals minus 3 i plus 9 j , v equals 5.8 i minus 7.1 j")[$bold(u) = − 3 bold(i) + 9 bold(j) , " " bold(v) = 5.8 bold(i) − 7.1 bold(j)$]


        
          #math.equation(block: true, alt: "2.8 i plus 1.9 j")[$2.8 bold(i) + 1.9 bold(j)$]


        
      
      
        #math.equation(block: true, alt: "u equals 6 i minus 8 j , v equals 23 i plus 42 j")[$bold(u) = 6 bold(i) − 8 bold(j) , " " bold(v) = 23 bold(i) + 42 bold(j)$]


      
      For Problems 31–38, find the coordinate form of the vector, where


      #math.equation(block: true, alt: "u equals 2 i plus 3 j , v equals minus 5 i plus 4 j , w equals minus 2 i minus 5 j , z equals 8 i minus 3 j")[$bold(u) = 2 bold(i) + 3 bold(j) , " " " " bold(v) = − 5 bold(i) + 4 bold(j) , " " " " bold(w) = − 2 bold(i) − 5 bold(j) , " " " " bold(z) = 8 bold(i) − 3 bold(j)$]
      
        #math.equation(block: true, alt: "u plus v")[$bold(u) + bold(v)$]


        
          #math.equation(block: true, alt: "minus 3 i plus 7 j")[$− 3 bold(i) + 7 bold(j)$]


        
      
      
        #math.equation(block: true, alt: "w minus z")[$bold(w) − bold(z)$]


      
      
        #math.equation(block: true, alt: "4 w")[$4 bold(w)$]


        
          #math.equation(block: true, alt: "minus 8 i minus 20 j")[$− 8 bold(i) − 20 bold(j)$]


        
      
      
        #math.equation(block: true, alt: "minus 3 v")[$− 3 bold(v)$]


      
      
        #math.equation(block: true, alt: "2 z minus u")[$2 bold(z) − bold(u)$]


        
          #math.equation(block: true, alt: "14 i minus 9 j")[$14 bold(i) − 9 bold(j)$]


        
      
      
        #math.equation(block: true, alt: "minus w plus 5 u")[$− bold(w) + 5 bold(u)$]


      
      
        #math.equation(block: true, alt: "3 v minus w plus 2 u")[$3 bold(v) − bold(w) + 2 bold(u)$]


        
          #math.equation(block: true, alt: "minus 9 i plus 23 j")[$− 9 bold(i) + 23 bold(j)$]


        
      
      
        #math.equation(block: true, alt: "z minus 2 open parenthesis v plus w close parenthesis")[$bold(z) − 2 ( bold(v) + bold(w) )$]


      
      For Problems 39–42, find a unit vector #math.equation(block: false, alt: "u")[$bold(u)$] in the same direction as the given vector.


      
        #math.equation(block: true, alt: "r equals minus 12 i plus 5 j")[$bold(r) = − 12 bold(i) + 5 bold(j)$]


        
          #math.equation(block: true, alt: "the fraction minus 12 over 13 i plus the fraction 5 over 13 j")[$display(frac(− 12, 13)) bold(i) + display(frac(5, 13)) bold(j)$]


        
      
      
        #math.equation(block: true, alt: "s equals 7 i minus 24 j")[$bold(s) = 7 bold(i) − 24 bold(j)$]


      
      
        #math.equation(block: true, alt: "t equals i minus j")[$bold(t) = bold(i) − bold(j)$]


        
          #math.equation(block: true, alt: "the fraction 1 over the square root of 2 i minus the fraction 1 over the square root of 2 j")[$display(frac(1, sqrt(2))) bold(i) − display(frac(1, sqrt(2))) bold(j)$]


        
      
      
        #math.equation(block: true, alt: "w equals minus 2 i minus 3 j")[$bold(w) = − 2 bold(i) − 3 bold(j)$]


      
      For Problems 43–46, find a vector #math.equation(block: false, alt: "v")[$bold(v)$] in the same direction as #math.equation(block: false, alt: "w")[$bold(w)$], but with the given length.


      
        #math.equation(block: true, alt: "w equals 8 i plus 15 j , ∥ v ∥ equals 51")[$bold(w) = 8 bold(i) + 15 bold(j) , " " ∥ bold(v) ∥ = 51$]


        
          #math.equation(block: true, alt: "24 i plus 45 j")[$24 bold(i) + 45 bold(j)$]


        
      
      
        #math.equation(block: true, alt: "w equals minus 20 i minus 21 j , ∥ v ∥ equals 58")[$bold(w) = − 20 bold(i) − 21 bold(j) , " " ∥ bold(v) ∥ = 58$]


      
      
        #math.equation(block: true, alt: "w equals minus 3 i plus j , ∥ v ∥ equals 4")[$bold(w) = − 3 bold(i) + bold(j) , " " ∥ bold(v) ∥ = 4$]


        
          #math.equation(block: true, alt: "the fraction minus 12 over the square root of 10 i plus the fraction 4 over the square root of 10 j")[$display(frac(− 12, sqrt(10))) bold(i) + display(frac(4, sqrt(10))) bold(j)$]


        
      
      
        #math.equation(block: true, alt: "w equals i minus 2 j , ∥ v ∥ equals 7")[$bold(w) = bold(i) − 2 bold(j) , " " ∥ bold(v) ∥ = 7$]


      
      For Problems 47–50,


      + Draw a diagram using arrows to represent the vectors.
+ Convert each vector to coordinate form.
+ Use the coordinate form to add or subtract the vectors.


      
        Find #math.equation(block: false, alt: "u plus v")[$bold(u) + bold(v)$], where #math.equation(block: false, alt: "u")[$bold(u)$] has magnitude 2.6 and direction #math.equation(block: false, alt: "θ equals 23 degrees")[$θ = 23^(∘)$], #math.equation(block: false, alt: "v")[$bold(v)$] has magnitude 5.8 and direction #math.equation(block: false, alt: "θ equals 223 degrees")[$θ = 223^(∘)$].


        
          + #figure(figph[vectors], alt: "vectors", caption: none)
+ #math.equation(block: false, alt: "u equals 2.393 i plus 1.016 j , v equals minus 4.242 i minus 3.956 j")[$bold(u) = 2.393 bold(i) + 1.016 bold(j) , " " " " bold(v) = − 4.242 bold(i) − 3.956 bold(j)$]
+ #math.equation(block: false, alt: "minus 1.849 i minus 2.940 j")[$− 1.849 bold(i) − 2.940 bold(j)$]


        
      
      
        Find #math.equation(block: false, alt: "u plus v")[$bold(u) + bold(v)$], where #math.equation(block: false, alt: "u")[$bold(u)$] has magnitude 50 and direction #math.equation(block: false, alt: "θ equals 173 degrees")[$θ = 173^(∘)$], #math.equation(block: false, alt: "v")[$bold(v)$] has magnitude 70 and direction #math.equation(block: false, alt: "θ equals 308 degrees")[$θ = 308^(∘)$].


      
      
        Find #math.equation(block: false, alt: "u minus v")[$bold(u) − bold(v)$], where #math.equation(block: false, alt: "u")[$bold(u)$] has magnitude 35 and direction #math.equation(block: false, alt: "θ equals 110 degrees")[$θ = 110^(∘)$], #math.equation(block: false, alt: "v")[$bold(v)$] has magnitude 60 and direction #math.equation(block: false, alt: "θ equals 165 degrees")[$θ = 165^(∘)$].


        
          + #figure(figph[vectors], alt: "vectors", caption: none)
+ #math.equation(block: false, alt: "u equals minus 11.97 i plus 32.889 j , v equals minus 57.955 i plus 15.529 j")[$bold(u) = − 11.97 bold(i) + 32.889 bold(j) , " " " " bold(v) = − 57.955 bold(i) + 15.529 bold(j)$]
+ #math.equation(block: false, alt: "45.98 i plus 17.36 j")[$45.98 bold(i) + 17.36 bold(j)$]


        
      
      
        Find #math.equation(block: false, alt: "u minus v")[$bold(u) − bold(v)$], where #math.equation(block: false, alt: "u")[$bold(u)$] has magnitude 12.4 and direction #math.equation(block: false, alt: "θ equals 250 degrees")[$θ = 250^(∘)$], #math.equation(block: false, alt: "v")[$bold(v)$] has magnitude 8.8 and direction #math.equation(block: false, alt: "θ equals 315 degrees")[$θ = 315^(∘)$].


      
      For Problems 51–56,


      + Make a sketch using vectors to illustrate the problem.
+ Use the coordinate form of the vectors to solve problem.


      
        The tornado displaced the trash bin to a spot 500 meters north and 800 meters east of its original position, and the flood later displaced the bin 2000 meters due south from there. How far and in what direction was the trash bin moved from it original position?


        
          + #figure(figph[vectors], alt: "vectors", caption: none)
+ 1700 m, #math.equation(block: false, alt: "28.1 degrees")[$28.1^(∘)$] east of south


        
      
      
        A radio-controlled model plane pointed due west with an airspeed of 15 miles per hour, but there was a crosswind from the north at a speed of 8 miles per hour. How fast and in what direction is the plane moving relative to the ground?


      
      
        Nimish flies 10 km in a direction #math.equation(block: false, alt: "175 degrees")[$175^(∘)$] north from east, then turns and flies an additional 12 km due west.  How far and in what direction is Nimish's final position relative to his starting point?


        
          + #figure(figph[vectors], alt: "vectors", caption: none)
+ 21.98 km, #math.equation(block: false, alt: "2.27 degrees")[$2.27^(∘)$] north of west


        
      
      
        Dena sails 500 yards due south, then turns and sails 350 yards in the direction #math.equation(block: false, alt: "300 degrees")[$300^(∘)$] from east. How far and in what direction is Dena's final position relative to her starting point?


      
      
        After leaving the airport, Kelly flew 30 miles at a heading #math.equation(block: false, alt: "30 degrees")[$30^(∘)$] east of north, then 50 miles #math.equation(block: false, alt: "70 degrees")[$70^(∘)$] east of north, and finally 12 miles #math.equation(block: false, alt: "20 degrees")[$20^(∘)$] south of east. What is her current position relative to the airport?


        
          + #figure(figph[vectors], alt: "vectors", caption: none)
+ 83 mi, #math.equation(block: false, alt: "62 degrees")[$62^(∘)$] east of north


        
      
      
        On a whale-watching trip, the SS Dolphin sailed 15 miles from port on a bearing of #math.equation(block: false, alt: "40 degrees")[$40^(∘)$], then 8 miles on a bearing of #math.equation(block: false, alt: "320 degrees")[$320^(∘)$], and then 4 miles on a bearing of #math.equation(block: false, alt: "250 degrees")[$250^(∘)$]. What is her current position relative to port?


      
      For Problems 57–60,


      + Find the resultant force.
+ Find the additional force needed for the system to be in equilibrium.


      
        #math.equation(block: true, alt: "F sub 1 equals minus 3 i plus j ,")[$bold(F)_(1) = − 3 bold(i) + bold(j) ,$] #math.equation(block: true, alt: "F sub 2 equals 5 i minus 2 j ,")[$" " bold(F)_(2) = 5 bold(i) − 2 bold(j) ,$] #math.equation(block: true, alt: "F sub 3 equals minus 6 i minus 4 j")[$" " bold(F)_(3) = − 6 bold(i) − 4 bold(j)$]


        
          + #math.equation(block: false, alt: "minus 4 i minus 5 j")[$− 4 bold(i) − 5 bold(j)$]
+ #math.equation(block: false, alt: "4 i plus 5 j")[$4 bold(i) + 5 bold(j)$]


        
      
      
        #math.equation(block: true, alt: "F sub 1 equals 10 i plus 4 j ,")[$bold(F)_(1) = 10 bold(i) + 4 bold(j) ,$] #math.equation(block: true, alt: "F sub 2 equals minus 12 i minus 9 j ,")[$" " bold(F)_(2) = − 12 bold(i) − 9 bold(j) ,$] #math.equation(block: true, alt: "F sub 3 equals minus 3 i plus 5 j")[$" " bold(F)_(3) = − 3 bold(i) + 5 bold(j)$]


      
      
        #figure(figph[vectors], alt: "vectors", caption: none)


        
          + #math.equation(block: false, alt: "i minus 3 j")[$bold(i) − 3 bold(j)$]
+ #math.equation(block: false, alt: "minus i plus 3 j")[$− bold(i) + 3 bold(j)$]


        
      
      
        #figure(figph[vectors], alt: "vectors", caption: none)


      
      
        + Find the magnitude of the vector#math.equation(block: false, alt: "v equals 6 i minus 8 j")[$bold(v) = 6 bold(i) − 8 bold(j)$], and the magnitude of the vector #math.equation(block: false, alt: "2 v equals 12 i minus 16 j")[$2 bold(v) = 12 bold(i) − 16 bold(j)$], and verify that #math.equation(block: false, alt: "∥ 2 v ∥ equals 2 ∥ v ∥")[$∥ 2 bold(v) ∥ = 2 ∥ bold(v) ∥$].
+ Let #math.equation(block: false, alt: "v equals a i plus b j")[$bold(v) = a bold(i) + b bold(j)$], and verify that #math.equation(block: false, alt: "∥ k v ∥ equals k ∥ v ∥")[$∥ k bold(v) ∥ = k ∥ bold(v) ∥$], for #math.equation(block: false, alt: "k greater than 0")[$k > 0$].


        
          + #math.equation(block: false, alt: "∥ v ∥ equals 10 , 2 ∥ v ∥ equals 20 equals 2 times 10")[$∥ bold(v) ∥ = 10 , " " 2 ∥ bold(v) ∥ = 20 = 2 ⋅ 10$]
+ #math.equation(block: false, alt: "∥ k v ∥ equals the square root of open parenthesis k a close parenthesis squared plus open parenthesis k b close parenthesis squared equals k the square root of a squared plus b squared")[$∥ k bold(v) ∥ = sqrt(( k a )^(2) + ( k b )^(2)) = k sqrt(a^(2) + b^(2))$]


        
      
      
        + Find the magnitude of the vector #math.equation(block: false, alt: "v equals 3 i plus 5 j")[$bold(v) = 3 bold(i) + 5 bold(j)$], and verify that the vector #math.equation(block: false, alt: "the fraction v over ∥ v ∥")[$display(frac(bold(v), ∥ bold(v) ∥))$] has magnitude 1.
+ Let #math.equation(block: false, alt: "v equals a i plus b j")[$bold(v) = a bold(i) + b bold(j)$], where #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are not both #math.equation(block: false, alt: "0")[$0$], and verify that #math.equation(block: false, alt: "the fraction v over ∥ v ∥")[$display(frac(bold(v), ∥ bold(v) ∥))$]  is a unit vector.