Vector Basics

A vector is a quantity with both a length (magnitude) and a direction, like a displacement, a velocity or a force. Draw it as an arrow; any arrow with the same length and direction is the same vector. A scalar is just a number.

Components

Put the tail at the origin and the tip lands at a point. Its coordinates are the vector's components:

$$\mathbf{v} = \langle v_1, v_2, v_3\rangle = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}$$

where $\mathbf{i}, \mathbf{j}, \mathbf{k}$ are the unit vectors along the $x$, $y$ and $z$ axes. The vector from $A(a_1, a_2, a_3)$ to $B(b_1, b_2, b_3)$ is $\overrightarrow{AB} = \langle b_1 - a_1, b_2 - a_2, b_3 - a_3\rangle$: tip minus tail.

Operations

Everything happens component by component:

$$\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2, u_3 + v_3\rangle, \qquad c\,\mathbf{v} = \langle cv_1, cv_2, cv_3\rangle$$

Geometrically, $\mathbf{u} + \mathbf{v}$ puts $\mathbf{v}$'s tail at $\mathbf{u}$'s tip; it's the diagonal of the parallelogram they span:

Multiplying by $c > 0$ stretches the vector; $c < 0$ also flips it around. Two vectors are parallel if one is a scalar multiple of the other.

Length

By Pythagoras (twice, in 3D):

$$\lvert\mathbf{v}\rvert = \sqrt{v_1^2 + v_2^2 + v_3^2}$$

A unit vector has length $1$. To get one pointing the same way as $\mathbf{v}$, divide by its length: $\frac{\mathbf{v}}{\lvert\mathbf{v}\rvert}$.

Distance and spheres

The distance between two points is the length of the vector between them. The sphere of radius $r$ centered at $(a, b, c)$ is every point at distance $r$ from the center:

$$(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$$

If a sphere's equation is expanded, complete the square in each variable to find its center and radius.