Vectors

A vector is a move: so many steps across, so many up. Watch two moves join tip to tail into one.

An arrow and a list

Say I tell you: “walk 3 steps east, then 1 step north.” That instruction is a vector. You can draw it as an arrow from where you start to where you end. You can also write it as a list of two numbers, $(3,1)$: steps across, then steps up.

The arrow and the list are the same thing seen two ways. On the grid at the top of the page, drag the tip of $\mathbf{v}$ and watch its readout change with it. Every place you can put the tip is one pair of numbers, and every pair of numbers is one place for the tip.

I write vectors in bold, $\mathbf{v}$, so you can tell them from ordinary numbers. The numbers in the list are the vector’s components: $v_1$ is the first (across) and $v_2$ the second (up). For $\mathbf{v}=(3,1)$, $v_1=3$ and $v_2=1$. I count components from 1.

An arrow does not belong to one spot. The move “3 east, 1 north” is the same move wherever you start it. Usually I draw a vector from the origin, the point $(0,0)$, but I may slide it anywhere without changing it.

Adding arrows: tip to tail

Make one move, then another. Walk $\mathbf{v}=(3,1)$, then from where you stopped walk $\mathbf{w}=(1,2)$. Altogether you went $3+1=4$ across and $1+2=3$ up. That total move is the sum $\mathbf{v}+\mathbf{w}$:

In the picture, you place the tail of $\mathbf{w}$ on the tip of $\mathbf{v}$, tip to tail. The sum is the single arrow from the start of the chain to its end. In the numbers, you add the components one by one.

The order does not matter. Walk $\mathbf{w}$ first, then $\mathbf{v}$, and you end at the same place. The two routes are the two sides of a parallelogram, and the sum is its diagonal:

Scaling an arrow

Walk $\mathbf{v}$ twice in a row and you have walked $(6,2)$: the same direction, twice as far. Multiplying a vector by an ordinary number $c$ multiplies every component by $c$:

An ordinary number used this way is called a scalar, because it scales the arrow. Four scalars tell the whole story, and the end of the clip at the top of the page plays them in turn:

• $2\mathbf{v}=(6,2)$: the same direction, twice as long.
• $\tfrac12\mathbf{v}=(1.5,\,0.5)$: half as long.
• $0\,\mathbf{v}=(0,0)$: no move at all. This is the zero vector, written $\mathbf{0}$, an arrow shrunk to a point.
• $-\mathbf{v}=(-3,-1)$: the same length, pointing the opposite way.

A scalar can stretch, shrink or flip an arrow. It can never turn it to a new direction off its line.

Subtracting: from one tip to the other

Subtraction is adding the flipped arrow: $\mathbf{w}-\mathbf{v}=\mathbf{w}+(-\mathbf{v})$. In numbers,

The picture is worth remembering. Draw $\mathbf{v}$ and $\mathbf{w}$ from the origin. The arrow from the tip of $\mathbf{v}$ to the tip of $\mathbf{w}$ is $\mathbf{w}-\mathbf{v}$. The check is one addition: start at the tip of $\mathbf{v}$, walk $\mathbf{w}-\mathbf{v}$, and you arrive at $\mathbf{v}+(\mathbf{w}-\mathbf{v})=\mathbf{w}$.

So $\mathbf{w}-\mathbf{v}$ answers the question “how do I get from here to there?” A game uses it to point a character at a target. A radar uses it for the gap between two aircraft.

More than three numbers

Nothing above needed the list to have two numbers. A point in a room has three: across, along and up. The rules are the same: add component by component, scale every component.

Now take eight pixels from one row of a photo. Each has a brightness, so the row is a list of eight numbers. Or take eight samples of a sound, eight numbers measured one after another. Each list is one vector, with eight components.

Watch the bars while the second list is added. Each bar only ever meets its own partner, the bar in the same place. That is the same rule as tip to tail, used eight times.

You can’t draw an arrow in eight dimensions, but you don’t need to. The list carries everything the arrow would. The set of all lists of $n$ real numbers is written $\mathbb{R}^n$ (“R n”). The plane is $\mathbb{R}^2$, the room is $\mathbb{R}^3$, and the row of pixels lives in $\mathbb{R}^8$.

Worked example

Take $\mathbf{v}=(3,1)$ and $\mathbf{w}=(1,2)$.

1. Sum. $\mathbf{v}+\mathbf{w}=(3+1,\;1+2)=(4,3)$.
2. Difference. $\mathbf{w}-\mathbf{v}=(1-3,\;2-1)=(-2,1)$: the arrow from the tip of $\mathbf{v}$, at $(3,1)$, to the tip of $\mathbf{w}$, at $(1,2)$.
3. A mixture. $2\mathbf{v}+\mathbf{w}=(6,2)+(1,2)=(7,4)$.
4. Flip. $-\mathbf{v}=(-3,-1)$.
5. Signals. A short pulse $\mathbf{x}=(0,\,0.5,\,1,\,0.5)$ plus a constant level $\mathbf{y}=(0.5,\,0.5,\,0.5,\,0.5)$ gives $\mathbf{x}+\mathbf{y}=(0.5,\,1,\,1.5,\,1)$: the same pulse, lifted by 0.5 everywhere.

Predict before you check: what does $0\cdot\mathbf{v}$ look like on the grid? It is a point at the origin, the zero vector.

Where you’ll meet this

Moving through wind. An aircraft flies at $(200,0)$ km/h through the air, 200 km/h due east. The wind blows at $(-30,40)$ km/h. Over the ground it moves at the sum, $(170,40)$ km/h: slower eastward, and drifting north.

Colours. A screen colour is a vector of three numbers, red, green and blue: $(255,128,0)$ is orange. Blending two colours mixes their vectors.

Games and graphics. A game stores every position and velocity as a vector. Each frame it adds velocity to position to move things along.

Sound. One second of audio sampled 8,000 times a second is one vector with 8,000 components, a single point in $\mathbb{R}^{8000}$. Linear algebra treats such long lists with the same rules you met here.

Reference card