#set document(title: "9.4 Chapter Summary and Review", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 9.4#h(0.6em)Chapter Summary and Review === Key Concepts + A quantity defined by both a magnitude (such as a distance) and a direction is called a #strong[vector]. + Two vectors are equal if they have the same length and direction; it does not matter where the vector starts. + The length of a vector #math.equation(block: false, alt: "v")[$bold(v)$] is called its #strong[magnitude], and is denoted by #math.equation(block: false, alt: "∥ v ∥")[$∥ bold(v) ∥$]. + The sum of two vectors #math.equation(block: false, alt: "u")[$bold(u)$] and #math.equation(block: false, alt: "v")[$bold(v)$] is a new vector, #math.equation(block: false, alt: "w")[$bold(w)$], starting at the tail of the first vector and ending at the head of the second vector. The sum is called the #strong[resultant vector]. + Addition of vectors is commutative. The rule for adding vectors is sometimes called the #strong[parallelogram rule]. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Operations on Vectors] + We can multiply a vector, #math.equation(block: false, alt: "v")[$bold(v)$], by a scalar, #math.equation(block: false, alt: "k")[$k$].+ If #math.equation(block: false, alt: "k greater than 0")[$k > 0$], the magnitude of #math.equation(block: false, alt: "k v")[$k bold(v)$] is #math.equation(block: false, alt: "k")[$k$] times the magnitude of #math.equation(block: false, alt: "v")[$bold(v)$]. The direction of #math.equation(block: false, alt: "k v")[$k bold(v)$] is the same as the direction of #math.equation(block: false, alt: "v")[$bold(v)$]. + If #math.equation(block: false, alt: "k less than 0")[$k < 0$], the direction of #math.equation(block: false, alt: "k v")[$k bold(v)$] is opposite the direction of #math.equation(block: false, alt: "v")[$bold(v)$]. + We can add two vectors #math.equation(block: false, alt: "v")[$bold(v)$] and#math.equation(block: false, alt: "w")[$bold(w)$] with the parallelogram rule. ] + Any vector can be written as the sum of its horizontal and vertical vector components, #math.equation(block: false, alt: "v sub x")[$bold(v)_(bold(x))$] and #math.equation(block: false, alt: "v sub y")[$bold(v)_(bold(y))$]. + The #strong[components] of a vector #math.equation(block: false, alt: "v")[$bold(v)$] whose direction is given by the angle #math.equation(block: false, alt: "θ")[$θ$] in standard position are the scalar quantities #math.equation(block: true, alt: "v sub x equals ∥ v ∥ cos ⁡ open parenthesis θ close parenthesis")[$v_(x) = ∥ bold(v) ∥ cos ( θ )$] #math.equation(block: true, alt: "v sub y equals ∥ v ∥ sin ⁡ open parenthesis θ close parenthesis")[$v_(y) = ∥ bold(v) ∥ sin ( θ )$] + The magnitude and direction of a vector with components and are given by #math.equation(block: true, alt: "∥ v ∥ equals the square root of open parenthesis v sub x close parenthesis squared plus open parenthesis v sub y close parenthesis squared and tan ⁡ open parenthesis θ close parenthesis equals the fraction v sub y over v sub x")[$∥ bold(v) ∥ = sqrt(( v_(x) )^(2) + ( v_(y) )^(2)) " " " " " " " " "and" " " " " " " " " tan ( θ ) = display(frac(v_(y), v_(x)))$] + To add two vectors using components, we can resolve each vector into its horizontal and vertical components, add the corresponding components, then compute the magnitude and direction of the resultant. + 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 ⁡ 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")[$a & = ∥ bold(v) ∥ cos ( θ ) & & & ∥ bold(v) ∥ & = sqrt(a^(2) + b^(2)) \ b & = ∥ bold(v) ∥ sin ( θ ) & & & tan ( θ ) & = display(frac(b, a))$] ] + #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 .")[$bold(w) = display(frac(k, ∥ bold(v) ∥)) bold(v) .$] ] + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Dot Product (Coordinate Formula)] The #strong[dot product] of two vectors #math.equation(block: false, alt: "v equals v sub 1 i plus v sub 2 j")[$bold(v) = v_(1) bold(i) + v_(2) bold(j)$] and #math.equation(block: false, alt: "w equals w sub 1 i plus w sub 2 j")[$bold(w) = w_(1) bold(i) + w_(2) bold(j)$] is the scalar #math.equation(block: true, alt: "v times w equals v sub 1 w sub 1 plus v sub 2 w sub 2")[$bold(v) ⋅ bold(w) = v_(1) w_(1) + v_(2) w_(2)$] ] + The dot product is a way of multiplying two vectors that depends on the angle between them. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Dot Product (Geometric Formula)] The #strong[dot product] of two vectors #math.equation(block: false, alt: "v")[$bold(v)$] and #math.equation(block: false, alt: "w")[$bold(w)$] is the scalar #math.equation(block: true, alt: "v times w equals ∥ v ∥ ∥ w ∥ cos ⁡ open parenthesis θ close parenthesis")[$bold(v) ⋅ bold(w) = ∥ bold(v) ∥ ∥ bold(w) ∥ cos ( θ )$]where #math.equation(block: false, alt: "θ")[$θ$] is the angle between the vectors. ] + The component of a vector #math.equation(block: false, alt: "w")[$bold(w)$] in the direction of vector #math.equation(block: false, alt: "v")[$bold(v)$] is the length of the vector projection of #math.equation(block: false, alt: "w")[$bold(w)$] onto #math.equation(block: false, alt: "w")[$bold(w)$]. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Component of a Vector] The component of #math.equation(block: false, alt: "w")[$bold(w)$] in the direction of #math.equation(block: false, alt: "v")[$bold(v)$] is the scalar #math.equation(block: true, alt: "comp sub v w equals the fraction v times w over ∥ v ∥")[$"comp"_(bold(v)) bold(w) = display(frac(bold(v) ⋅ bold(w), ∥ bold(v) ∥))$] ] + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Angle Between Two Vectors] The angle #math.equation(block: false, alt: "θ")[$θ$] between two vectors #math.equation(block: false, alt: "v")[$bold(v)$] and #math.equation(block: false, alt: "w")[$bold(w)$] is given by #math.equation(block: true, alt: "cos ⁡ open parenthesis θ close parenthesis equals the fraction v times w over ∥ v ∥ ∥ w ∥")[$cos ( θ ) = display(frac(bold(v) ⋅ bold(w), ∥ bold(v) ∥ ∥ bold(w) ∥))$] ] + Two vectors #math.equation(block: false, alt: "v")[$bold(v)$] and #math.equation(block: false, alt: "w")[$bold(w)$] are orthogonal if #math.equation(block: false, alt: "v times w equals 0")[$bold(v) ⋅ bold(w) = 0$] === Review Problems For Problems 1–4, sketch an arrow to represent the vector. Find its components in the directions north and east. A hawk is flying at a speed of 20 mph in the direction #math.equation(block: false, alt: "65 degrees")[$65^(∘)$] west of north. #figure(figph[vector], alt: "vector", caption: none) #math.equation(block: false, alt: "v sub N equals 8.45")[$v_(N) = 8.45$] mph, #math.equation(block: false, alt: "v sub E equals minus 18.13")[$v_(E) = − 18.13$] mph The island is located 36 miles from port on a bearing of #math.equation(block: false, alt: "160 degrees")[$160^(∘)$]. The tractor pulls with a force of 1200 pounds in the direction #math.equation(block: false, alt: "20 degrees")[$20^(∘)$] west of south. #figure(figph[vector], alt: "vector", caption: none) #math.equation(block: false, alt: "v sub N equals minus 1127.63")[$v_(N) = − 1127.63$] lbs, #math.equation(block: false, alt: "v sub E equals minus 410.42")[$v_(E) = − 410.42$] lbs The current runs southeast at a speed of 4 mph. For Problems 5–6, find the magnitude and direction of the vector. #math.equation(block: true, alt: "A sub x equals minus 6 , A sub y equals minus 9")[$A_(x) = − 6 , " " A_(y) = − 9$] #math.equation(block: true, alt: "∥ A ∥ equals 10.9 , θ equals 236.3 degrees")[$∥ bold(A) ∥ = 10.9 , " " θ = 236.3^(∘)$] #math.equation(block: true, alt: "w sub x equals 15.2 , w sub y equals minus 8.6")[$w_(x) = 15.2 , " " w_(y) = − 8.6$] For Problems 7–8, find the coordinate form of the vector. #math.equation(block: true, alt: "∥ v ∥ equals 2 , θ equals 300 degrees")[$∥ bold(v) ∥ = 2 , " " θ = 300^(∘)$] #math.equation(block: true, alt: "i minus the square root of 3 j")[$bold(i) − sqrt(3) bold(j)$] #math.equation(block: true, alt: "∥ v ∥ equals 10 , θ equals 225 degrees")[$∥ bold(v) ∥ = 10 , " " θ = 225^(∘)$] For Problems 9–12, + sketch the displacement vector and give its coordinate form, + find the magnitude and direction of the vector. The displacement vector from #math.equation(block: false, alt: "open parenthesis minus 8 , minus 4 close parenthesis")[$( − 8 , − 4 )$] to #math.equation(block: false, alt: "open parenthesis 7 , minus 1 close parenthesis")[$( 7 , − 1 )$] + #figure(figph[vector], alt: "vector", caption: none) #math.equation(block: true, alt: "15 i plus 3 j")[$15 bold(i) + 3 bold(j)$] + #math.equation(block: false, alt: "∥ v ∥ equals 15.3 , θ equals 11.3 degrees")[$∥ bold(v) ∥ = 15.3 , " " θ = 11.3^(∘)$] The displacement vector from #math.equation(block: false, alt: "open parenthesis 5 , 35 close parenthesis")[$( 5 , 35 )$] to #math.equation(block: false, alt: "open parenthesis minus 10 , 15 close parenthesis")[$( − 10 , 15 )$] This morning we began hiking from our camp 4 miles east and 2 miles south of the lodge, and this evening we are 6 miles east and 8 miles south of the lodge. + #figure(figph[vector], alt: "vector", caption: none) #math.equation(block: true, alt: "2 i minus 6 j")[$2 bold(i) − 6 bold(j)$] + #math.equation(block: false, alt: "∥ v ∥ equals 6.3 mi , θ equals 288.4 degrees")[$∥ bold(v) ∥ = 6.3 " " "mi" , " " θ = 288.4^(∘)$] The tunnel should start 100 meters east and 400 meters north of the survey point, and should end 500 meters west and 150 meters north of the survey point. For Problems 13–16, + sketch the given vectors, + calculate the magnitude and direction of the resultant vector. A fire crew is located 2 kilometers due west of the fire station. The station reports a new hot spot 6 kilometers away in the direction #math.equation(block: false, alt: "50 degrees")[$50^(∘)$] east of north. How far is the hot spot from the fire crew, and in what direction? + #figure(figph[vectors], alt: "vectors", caption: none) + #math.equation(block: false, alt: "7.64 km , θ equals 30.31 degrees")[$7.64 " " "km" , " " θ = 30.31^(∘)$] A helicopter has just delivered a patient to the hospital located 15 miles northwest of the heliport. The pilot gets a call to pick up a passenger located 18 miles from the heliport on a bearing of #math.equation(block: false, alt: "200 degrees")[$200^(∘)$]. How far is the passenger from the helicopter, and in what direction? Red Rock is located at #math.equation(block: false, alt: "4.2 i plus 2.8 j")[$4.2 bold(i) + 2.8 bold(j)$] from the town of Dry Gulch, measured in miles, and Skull Point is located at #math.equation(block: false, alt: "minus 3.5 i plus 6.3 j")[$− 3.5 bold(i) + 6.3 bold(j)$] from Dry Gulch. How far is it from Red Rock to Skull Point, and in what direction? + #figure(figph[vectors], alt: "vectors", caption: none) + #math.equation(block: false, alt: "8.46 mi , θ equals 155.6 degrees")[$8.46 " " "mi" , " " θ = 155.6^(∘)$] A coast guard cutter is located 7 miles south and 5 miles west of port when it gets a distress call from a sailboat that reports its location as 1 mile north and 5 miles east of port. How far is it from the cutter to the sailboat, and in what direcion? For Problems 17–18, + find the horizontal and vertical components of the forces, + use the components to calculate the resultant force. #figure(figph[vectors], alt: "vectors", caption: none) + #math.equation(block: false, alt: "F sub 1 equals minus 200 i , F sub 2 equals minus 60 the square root of 2 i minus 60 the square root of 2 j , F sub 3 equals 50 the square root of 3 i plus 50 j ,")[$bold(F)_(1) = − 200 bold(i) , " " bold(F)_(2) = − 60 sqrt(2) bold(i) − 60 sqrt(2) bold(j) , " " bold(F)_(3) = 50 sqrt(3) bold(i) + 50 bold(j) ,$] #math.equation(block: false, alt: "F sub 4 equals minus 125 i plus 125 the square root of 3 j")[$bold(F)_(4) = − 125 bold(i) + 125 sqrt(3) bold(j)$] + #math.equation(block: false, alt: "minus 73.25 i plus 181.65 j")[$− 73.25 bold(i) + 181.65 bold(j)$] #figure(figph[vectors], alt: "vectors", caption: none) For Problems 19–22, find the vector, where #math.equation(block: true, alt: "u equals 4 i plus 2 j , v equals minus 3 i minus j , w equals 2 i minus 3 j")[$bold(u) = 4 bold(i) + 2 bold(j) , " " " " " " bold(v) = − 3 bold(i) − bold(j) , " " " " " " bold(w) = 2 bold(i) − 3 bold(j)$] #math.equation(block: true, alt: "u minus 3 v")[$bold(u) − 3 bold(v)$] #math.equation(block: true, alt: "13 i plus 5 j")[$13 bold(i) + 5 bold(j)$] #math.equation(block: true, alt: "v minus 2 open parenthesis u minus w close parenthesis")[$bold(v) − 2 ( bold(u) − bold(w) )$] #math.equation(block: true, alt: "3 open parenthesis v plus w close parenthesis minus u")[$3 ( bold(v) + bold(w) ) − bold(u)$] #math.equation(block: true, alt: "minus 7 i minus 14 j")[$− 7 bold(i) − 14 bold(j)$] #math.equation(block: true, alt: "2 u minus 3 w minus v")[$2 bold(u) − 3 bold(w) − bold(v)$] For Problems 23–26, find the vector described. The unit vector in the same direction as #math.equation(block: false, alt: "2 i plus 3 j")[$2 bold(i) + 3 bold(j)$]. #math.equation(block: true, alt: "the fraction 2 over the square root of 13 i plus the fraction 3 over the square root of 13 j")[$display(frac(2, sqrt(13))) bold(i) + display(frac(3, sqrt(13))) bold(j)$] The unit vector in the same direction as #math.equation(block: false, alt: "5 i plus 12 j")[$5 bold(i) + 12 bold(j)$]. The vector of length 3 in the same direction as #math.equation(block: false, alt: "minus 2 i minus 5 j")[$− 2 bold(i) − 5 bold(j)$]. #math.equation(block: true, alt: "the fraction minus 6 over the square root of 29 i minus the fraction 15 over the square root of 29 j")[$display(frac(− 6, sqrt(29))) bold(i) − display(frac(15, sqrt(29))) bold(j)$] The vector of magnitude 6 in the same direction as #math.equation(block: false, alt: "minus 3 i plus 2 j")[$− 3 bold(i) + 2 bold(j)$] For Problems 27–28, find the component of #math.equation(block: false, alt: "w")[$bold(w)$] in the direction of #math.equation(block: false, alt: "v")[$bold(v)$]. #math.equation(block: true, alt: "v equals minus 6 i minus j , w equals 4 i minus 3 j")[$bold(v) = − 6 bold(i) − bold(j) , " " " " bold(w) = 4 bold(i) − 3 bold(j)$] #math.equation(block: true, alt: "minus 3.45")[$− 3.45$] #math.equation(block: true, alt: "v equals minus 2 i plus j , w equals i minus 2 j")[$bold(v) = − 2 bold(i) + bold(j) , " " " " bold(w) = bold(i) − 2 bold(j)$] For Problems 29–30, compute the dot product #math.equation(block: false, alt: "u times v")[$bold(u) ⋅ bold(v)$]. #math.equation(block: true, alt: "u equals 3.8 i plus 4.8 j ,")[$bold(u) = 3.8 bold(i) + 4.8 bold(j) ,$] #math.equation(block: true, alt: "v equals minus 9.2 i plus 5.6 j")[$" " bold(v) = − 9.2 bold(i) + 5.6 bold(j)$] #math.equation(block: true, alt: "minus 8.08")[$− 8.08$] #math.equation(block: true, alt: "u equals minus 27 i plus 35 j ,")[$bold(u) = − 27 bold(i) + 35 bold(j) ,$] #math.equation(block: true, alt: "v equals minus 16 i minus 24 j")[$" " bold(v) = − 16 bold(i) − 24 bold(j)$] For Problems 31–32, find the angle between the vectors. #math.equation(block: true, alt: "v equals minus 4 i minus 3 j , w equals 4 i minus 3 j")[$bold(v) = − 4 bold(i) − 3 bold(j) , " " " " bold(w) = 4 bold(i) − 3 bold(j)$] #math.equation(block: true, alt: "106.26 degrees")[$106.26^(∘)$] #math.equation(block: true, alt: "v equals 8 i minus 2 j ,")[$bold(v) = 8 bold(i) − 2 bold(j) ,$] #math.equation(block: true, alt: "w equals minus 5 i minus j")[$" " bold(w) = − 5 bold(i) − bold(j)$]