A plain number — like 7, or -3.5 — is called a scalar. It carries a single piece of information: a size. That's perfect for a temperature or a price, but useless for a quantity that also points somewhere. If I tell you a plane flew 600 km, you still don't know where it ended up. You also need a direction.

A vector is a quantity with both a magnitude (a size) and a direction. We draw it as an arrow: the length is the magnitude, and the way it points is the direction. Velocity, force, displacement and acceleration are all vectors — which is exactly why physics is so fond of arrows.

An arrow you can steer

Below is a single vector drawn from the origin. One slider sets its magnitude (how long it is) and the other sets its direction (the angle it points, measured anticlockwise from the positive x-axis). Change them and watch the arrow obey. Notice that the two numbers — length and angle — are enough to pin the arrow down completely.

Writing vectors down

To tell a vector apart from a scalar in writing, we use a little arrow or bold type:

The magnitude — the length of the arrow — is written with bars, like an absolute value: \lvert \vec{v} \rvert or \lVert \vec{v} \rVert. A vector with the same direction as \vec{v} but twice as long is a different vector; one pointing the opposite way is different again. Only when both magnitude and direction match are two vectors equal — and that's true even if they start from different places, because a vector stores no "home", just a length and a heading.