Vectors: The Basics

A vector is simply an amount with a direction attached: "5 km northeast" instead of just "5 km." The name comes from the Latin for "carrier," since a vector carries you from one point to another. It's how we describe anything that has a size and points somewhere, like a velocity or a force. The notation $\langle a_1, a_2, a_3\rangle$ looks formal, but it's just three numbers: $a_1$ (how far east), $a_2$ (how far north) and $a_3$ (how far up), and the length $\|\vec v\|$ is the plain old Pythagorean theorem. That's really all a vector is, so bear with the symbols.

In single-variable calculus a number was enough: how far, how fast. Once you're moving around in space, you also need to say which way. That's what a vector is for. It packs a size and a direction into one object, which is why displacement, velocity, force and (later in this course) the gradient are all vectors.

Components. To get the vector from $P$ to $Q$, take "end minus start" in each coordinate. Notice that only the difference matters, not where you started. So two arrows with the same components are the same vector, wherever they're drawn. When you do want to pin a vector to a point, use its position vector, the arrow from the origin to that point.

Magnitude is the length. It's just the Pythagorean theorem: $\|\langle 3, 4\rangle\| = 5$. In 3D you use it twice, once across the floor and once going up, which is where the third square under the root comes from.

Unit vectors have length 1, so all they carry is direction. You get one from any nonzero vector by dividing by its length; that shrinks or stretches it to length 1 without turning it. The standard ones point along the axes:

so $\langle 2, -1, 4\rangle = 2\vec i - \vec j + 4\vec k$. The components and the $\vec i, \vec j, \vec k$ form say exactly the same thing.

Game engines store every position, velocity and force as a vector; normalizing (making unit vectors) is how they get pure directions for lighting and movement.