0.4 Scalar Multiples: Stretch, Shrink, Reverse

The second operation on vectors is multiplication by a number. Doubling a vector doubles its length and keeps its direction; halving it halves the length. A number used this way is called a scalar, because scaling is exactly what it does, and the product $k\mathbf{v}$ is a scalar multiple of $\mathbf{v}$. Two cases need a moment's thought. Multiplying by $-1$ gives $-\mathbf{v}$, the vector of the same length pointing the opposite way — the additive inverse of the last section — and any negative scalar both scales and reverses. Multiplying by $0$ gives the zero vector, whatever $\mathbf{v}$ was.

The figure keeps a fixed vector $\mathbf{v} = (2, 1)$ and draws $k\mathbf{v}$ on a slider that runs through negative values, over the dashed line of all multiples.

Interactive figureScalar multiples of a vectorDrag the the scalar slider from -3 to 3.

▶ Explore interactively

XYZ Graph · viewer build 5edf91b

Explore the figure

1. At the starting value $k = 1.5$, the blue arrow ends at $(3, 1.5)$: each component of $\mathbf{v}$ multiplied by $1.5$. It points the way $\mathbf{v}$ points and is one and a half times as long.
2. Drag $k$ to $0.5$. The arrow is half of $\mathbf{v}$, ending at the midpoint $(1, 0.5)$ of the red arrow. Now drag $k$ to $3$: the arrow triples, to $(6, 3)$.
3. Drag $k$ to $0$. The blue arrow shrinks to nothing: $0\mathbf{v}$ is the zero vector, with no length to draw and no direction to point.
4. Drag $k$ to $-1$. The arrow reappears on the other side of the origin, ending at $(-2, -1)$, the same length as $\mathbf{v}$ and pointing the opposite way. That is $-\mathbf{v}$; at $k = -2$ it is twice as long and still reversed.
5. Sweep $k$ across its whole range and watch the blue tip. It never leaves the dashed line. The multiples of one nonzero vector fill exactly one line through the origin, and every point of that line is one of them.

Why components scale

The figure shows components scaling with the arrow, and a similar-triangles argument says why. The arrow $\mathbf{v} = (v_1, v_2)$ is the hypotenuse of a right triangle with legs $v_1$ and $v_2$ along the axes. Stretching the hypotenuse by a factor $k > 0$ from the origin produces a similar triangle, with every side stretched by $k$; its legs are $kv_1$ and $kv_2$, and those are the components of the stretched arrow. A negative $k$ reflects the triangle through the origin as well, which negates both legs. The same argument runs in space with three legs, and in $\mathbb{R}^n$ we take the component rule as the definition:

$$k\mathbf{v} = (kv_1, \; kv_2, \; \dots, \; kv_n)$$

Length behaves as the picture promised. Squaring each component multiplies it by $k^2$, so $|k\mathbf{v}|^2 = k^2 |\mathbf{v}|^2$ and

$$|k\mathbf{v}| = |k| \, |\mathbf{v}|$$

with the absolute value on $k$ because a length cannot be negative: $-2\mathbf{v}$ is twice as long as $\mathbf{v}$, not minus twice as long.

Two nonzero vectors that are scalar multiples of each other lie along one line, and we call them parallel — which includes the case of pointing opposite ways. Conversely, if $\mathbf{w}$ lies on the line through $\mathbf{v}$ then $\mathbf{w} = k\mathbf{v}$ for the unique $k$ that scales $\mathbf{v}$ to the right length and sign. Testing for parallelism is therefore a component question: is there one number $k$ that works for every component at once?

Scalar multiplication brings the last four rules of vector arithmetic, again because each is a rule of ordinary numbers applied in every component. For all vectors $\mathbf{u}$, $\mathbf{v}$ and scalars $k$, $p$:

• $k(\mathbf{u} + \mathbf{v}) = k\mathbf{u} + k\mathbf{v}$ and $(k + p)\mathbf{u} = k\mathbf{u} + p\mathbf{u}$: scaling distributes over both kinds of addition.
• $k(p\mathbf{u}) = (kp)\mathbf{u}$ and $1\mathbf{u} = \mathbf{u}$.

The first is worth writing out once. The $i$th component of $k(\mathbf{u} + \mathbf{v})$ is $k(u_i + v_i) = ku_i + kv_i$, which is the $i$th component of $k\mathbf{u} + k\mathbf{v}$; and two vectors whose components all agree are the same vector. Together with the four rules for addition, these eight are everything the rest of the book uses about the two operations. That is worth saying precisely, because it is what Chapter 2 leans on: every argument there about spans, subspaces and independence is built from these eight rules and nothing else about what a vector is. Mathematicians take the same eight as the definition of a vector space, which is what lets the subject reach objects that are not lists of numbers at all — functions, polynomials, matrices. This book stays inside $\mathbb{R}^n$, so that generalization is a door we point at rather than walk through.

An original work of XYZ Homework, built around interactive XYZ 3D figures. Its chapter sequence is aligned to Interactive Linear Algebra (Margalit & Rabinoff, Georgia Tech, GNU FDL); this work is original, copies nothing from it, and is not affiliated with or endorsed by its authors. License: CC-BY-NC-SA-4.0.