What Is a Vector?

A weather forecaster doesn't just say "the wind is 20 km/h." She says "20 km/h from the northwest." A pilot doesn't just care that the plane is doing 900 km/h — she needs to know 900 km/h towards which airport. A tug-of-war team doesn't just care how hard the other side is pulling — they care which way the rope is being pulled. Over and over, some quantities refuse to be pinned down by a single number. They need a size and a direction at the same time.

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 — it could be 600 km north, or 600 km straight down into an unfortunate ocean. 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. In fact, once you start looking for them, vectors turn out to be one of the single most useful ideas in all of applied mathematics: they describe how a ball flies, how a spacecraft steers, how light bends, and how a video-game character moves across the screen.

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. Nothing else about it is free to change.

Free vectors: the same arrow, many homes

Here is the strange and useful part: a vector has no fixed starting point. Draw an arrow 3 units long, pointing due east, starting at the origin. Now draw another arrow 3 units long, pointing due east, starting somewhere else entirely — over in the corner of the page. Even though they live in different places, they are considered exactly the same vector. All a vector remembers is "how far" and "which way" — it has amnesia about where it began.

This matters because it lets us slide vectors around freely to compare them, add them, or carry them tip-to-tail without changing what they mean — an idea you'll lean on heavily once you start adding vectors together.

It's tempting to think an ordered pair like (3, 4) is simply a location — a dot on the page. But the very same pair of numbers can mean two completely different things depending on context:

A point is a place. A vector is a move. Keep that distinction straight and a great many later confusions simply vanish.

Writing vectors down

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

\vec{v} \quad\text{or}\quad \mathbf{v}.

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, exactly as the previous section showed.

There's also a second, very common way to write a vector down: as an ordered pair of numbers — its components. If an arrow's horizontal stretch is 3 and its vertical stretch is 4, we write \vec{v} = (3, 4). This is exactly the same information as "magnitude 5, pointing up and to the right" — just packaged differently. You'll meet this coordinate form properly very soon; for now, just notice that "arrow", "magnitude and direction", and "a pair of numbers" are three ways of saying the same thing.

Worked example: vector, or just a scalar?

A good test for "is this a vector?" is to ask: does a direction make it more meaningful, or does it not even make sense? Let's run three everyday pairs through that test.

Speed vs. velocity. "The car is going 60 km/h" is a scalar — just a size. "The car is going 60 km/h due north" is a vector — a size and a direction. Same number, but velocity carries extra information that speed throws away.

Mass vs. weight. "This bag of flour has a mass of 2 kg" is a scalar: it's simply an amount of stuff, and asking "which way is the mass pointing?" is nonsense. But "gravity pulls on the flour with a force of about 20 newtons, straight down" is a vector — weight is a force, and force always acts in some direction.

Temperature vs. wind. "It's 15°C outside" is a scalar — one number describes it completely. "The wind is blowing 15 km/h from the southwest" is a vector — the direction is part of what makes the description useful; a sailor who only heard "15 km/h" with no direction couldn't set her sails correctly.

Notice the pattern: whenever "which way?" is a sensible follow-up question, you're looking at a vector. Whenever "which way?" makes no sense at all, you're looking at a scalar. A few more to try the test on yourself: distance travelled along a winding road (a scalar — just a length) versus displacement, the straight-line arrow from start to finish (a vector — it has a direction, and it's usually shorter than the distance!); or acceleration, the rate at which velocity changes (a vector, because "speeding up" and "slowing down while turning" both involve a direction) versus a car's fuel economy in litres per 100 km (a scalar, with no direction attached at all).

Worked example: reading components off a diagram

Suppose an arrow starts at the origin and its tip lands on the point (4, 3), as drawn below. To find its components, just read off how far the tip moved horizontally and vertically: 4 across, 3 up. So \vec{v} = (4, 3).

To find the magnitude — the actual length of the arrow — treat the components as the two shorter sides of a right triangle and use Pythagoras's theorem:

\lvert \vec{v} \rvert = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5.

So this arrow is 5 units long. Components and magnitude are two different questions about the very same arrow — "how far in each direction?" versus "how far in total?" — and Pythagoras is the bridge between them.

This two-way relationship — components in, magnitude and direction out, and back again — is worth practising until it feels automatic, because you'll use it constantly: to check a rocket's speed from its horizontal and vertical velocity components, to find how far a hiker really is from base camp after wandering across a map, or simply to check whether an arrow you've drawn actually has the length you intended.

The word vector comes from the Latin vehere, "to carry." A mathematical vector "carries" a point from its tail to its tip — it's literally a carrier of displacement.

Biologists borrowed the very same word for a completely different reason: a disease vector is an organism — a mosquito, a tick, a flea — that carries a pathogen from one host to another, the way malaria hitches a ride on a mosquito from person to person. Different field, same root idea: something that carries something else from one place to another. Next time you hear "vector" on the news during flu season, you'll know exactly where the word came from.

Every time a character jumps, a ball bounces, or a spaceship banks left in a video game, the game's physics engine is quietly doing vector arithmetic behind the scenes. Position, velocity, acceleration, and every force pushing or pulling on an object — gravity, thrust, a collision bump — are all stored as vectors, usually a tiny list of two or three numbers per object, updated dozens of times a second.

This is also exactly how sailors and pilots have worked for centuries, long before computers existed: a ship's captain plots her intended course as one vector, notices the current pushing her sideways is a second vector, and combines the two to predict — and correct for — where she'll actually end up. Whether it's pixels on a screen or a real ship on real water, the underlying trick is the same: represent "how far and which way" as a single arrow, and let the arithmetic of vectors do the rest.

See it explained