Adding, Subtracting and Scaling Vectors

Adding end to end

Walk $4$ blocks east and $1$ north, then $1$ east and $3$ north. Where have you ended up, compared with where you started?

Each walk is a vector: $\mathbf{v} = \langle 4, 1\rangle$ and $\mathbf{w} = \langle 1, 3\rangle$. Doing one after the other means starting $\mathbf{w}$ where $\mathbf{v}$ ends. The total trip, the sum, runs from the first start to the last finish.

In components, the moves across add and the moves up add:

$$\langle 4, 1\rangle + \langle 1, 3\rangle = \langle 5, 4\rangle$$

The dashed arrows show the same sum reached the other way round, $\mathbf{w}$ first. The two routes form a parallelogram, and the sum is its diagonal. That is the parallelogram rule, and it shows the order does not matter:

$$\mathbf{v} + \mathbf{w} = \mathbf{w} + \mathbf{v}$$

Why the length of a sum is not the sum of the lengths

$\mathbf{v}$ is about $4.12$ long and $\mathbf{w}$ about $3.16$. Their sum, $\langle 5, 4\rangle$, is about $6.40$ long — not $7.28$.

The three arrows form a triangle, and one side of a triangle is always shorter than the other two combined. Going straight from start to finish is shorter than detouring through the corner.

$$\lVert\mathbf{v} + \mathbf{w}\rVert \le \lVert\mathbf{v}\rVert + \lVert\mathbf{w}\rVert$$

The two sides match only when the arrows point the same way, and the triangle flattens into a line. Pushes in different directions partly work against each other, which is exactly what the shorter sum records.

Adding vectors given by magnitude and direction

When vectors come as a size and a direction, convert each to components, add, then convert back.

A plane heads due north at $300$ miles per hour. A wind blows due east at $50$ miles per hour. What is the plane’s actual velocity?

$$\langle 0, 300\rangle + \langle 50, 0\rangle = \langle 50, 300\rangle$$ $$\text{speed} = \sqrt{50^2 + 300^2} \approx 304.1 \text{ mph}$$

The direction is east of north by the angle whose tangent is $\tfrac{50}{300}$: about $9.46°$. The wind both speeds the plane slightly and pushes it off course.

Subtraction reverses an arrow

$-\mathbf{w}$ has the same length as $\mathbf{w}$ and points the opposite way. Subtracting is adding the reverse:

$$\mathbf{v} - \mathbf{w} = \mathbf{v} + (-\mathbf{w})$$

In components the minus signs do it: $\langle 4, 1\rangle - \langle 1, 3\rangle = \langle 3, -2\rangle$.

On a diagram, with both arrows starting together, $\mathbf{v} - \mathbf{w}$ is the arrow from the tip of $\mathbf{w}$ to the tip of $\mathbf{v}$ — because $\mathbf{w}$ plus that arrow gets you to $\mathbf{v}$.

Scaling a vector

Multiplying a vector by a number $c$, a scalar, multiplies each component:

$$c\langle a, b\rangle = \langle ca, cb\rangle$$

The length is multiplied by $\lvert c\rvert$:

$$\lVert c\mathbf{v}\rVert = \lvert c\rvert\,\lVert\mathbf{v}\rVert$$

The direction stays the same when $c > 0$ and reverses when $c < 0$.

Worked examples

Common mistakes

Practice problems