Vectors and geometry

$\mathbb{R}^n$ is the set of $n$-tuples $\mathbf{v} = (v_1, \dots, v_n)$, written as columns. Vectors are added and scaled componentwise:

Geometrically $\mathbf{v}$ is an arrow (displacement); $\mathbf{u} + \mathbf{v}$ is "$\mathbf{u}$ then $\mathbf{v}$" (parallelogram rule). The standard basis is $\mathbf{e}_1, \dots, \mathbf{e}_n$, with $\mathbf{v} = \sum v_i\mathbf{e}_i$. A linear combination of $\mathbf{v}_1, \dots, \mathbf{v}_k$ is $\sum\lambda_i\mathbf{v}_i$.

$\mathbf{u}\cdot\mathbf{v} = \sum_{i=1}^nu_iv_i$ and $\|\mathbf{v}\| = \sqrt{\mathbf{v}\cdot\mathbf{v}}$. Properties: $\mathbf{u}\cdot\mathbf{v} = \mathbf{v}\cdot\mathbf{u}$; linear in each argument; $\mathbf{v}\cdot\mathbf{v}\ge0$ with equality iff $\mathbf{v} = \mathbf{0}$; $\|\lambda\mathbf{v}\| = |\lambda|\|\mathbf{v}\|$. A unit vector has length $1$; $\frac{\mathbf{v}}{\|\mathbf{v}\|}$ is the unit vector in the direction of $\mathbf{v}\ne\mathbf{0}$. The distance between points is $\|\mathbf{u} - \mathbf{v}\|$.

For all $\mathbf{u}, \mathbf{v}\in\mathbb{R}^n$: $|\mathbf{u}\cdot\mathbf{v}|\le\|\mathbf{u}\|\|\mathbf{v}\|$, with equality iff $\mathbf{u}, \mathbf{v}$ are parallel (one is a multiple of the other). In coordinates: $\big(\sum u_iv_i\big)^2\le\big(\sum u_i^2\big)\big(\sum v_i^2\big)$.

$\|\mathbf{u} + \mathbf{v}\|\le\|\mathbf{u}\| + \|\mathbf{v}\|$. For non-zero $\mathbf{u}, \mathbf{v}$ the angle $\theta\in[0, \pi]$ between them is defined by $\cos\theta = \frac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}$. They are orthogonal ($\mathbf{u}\perp\mathbf{v}$) iff $\mathbf{u}\cdot\mathbf{v} = 0$, and then $\|\mathbf{u} + \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2$ (Pythagoras).

For $\mathbf{a}\ne\mathbf{0}$, the projection of $\mathbf{v}$ onto $\mathbf{a}$ is $\operatorname{proj}_{\mathbf{a}}\mathbf{v} = \frac{\mathbf{v}\cdot\mathbf{a}}{\mathbf{a}\cdot\mathbf{a}}\mathbf{a}$, the "shadow" of $\mathbf{v}$ on the line through $\mathbf{a}$.

$\mathbf{v} = \operatorname{proj}_{\mathbf{a}}\mathbf{v} + \mathbf{w}$ with $\mathbf{w}\perp\mathbf{a}$, and $\operatorname{proj}_{\mathbf{a}}\mathbf{v}$ is the multiple of $\mathbf{a}$ closest to $\mathbf{v}$.

For $\mathbf{u}, \mathbf{v}\in\mathbb{R}^3$, $\mathbf{u}\times\mathbf{v} = (u_2v_3 - u_3v_2,\ u_3v_1 - u_1v_3,\ u_1v_2 - u_2v_1)$. Properties: $\mathbf{v}\times\mathbf{u} = -\mathbf{u}\times\mathbf{v}$; bilinear; $\mathbf{u}\times\mathbf{v}$ is orthogonal to both $\mathbf{u}$ and $\mathbf{v}$; $\|\mathbf{u}\times\mathbf{v}\| = \|\mathbf{u}\|\|\mathbf{v}\|\sin\theta$, the area of the parallelogram they span; $\mathbf{u}\times\mathbf{v} = \mathbf{0}$ iff parallel. Direction by the right-hand rule: $\mathbf{e}_1\times\mathbf{e}_2 = \mathbf{e}_3$. Not associative.

$[\mathbf{a}, \mathbf{b}, \mathbf{c}] = \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$ equals the determinant of the matrix with rows $\mathbf{a}, \mathbf{b}, \mathbf{c}$. Its absolute value is the volume of the parallelepiped they span; it is $0$ iff the three vectors are coplanar. It is unchanged by cyclic permutation and changes sign under a swap.

The line through $\mathbf{a}$ with direction $\mathbf{d}\ne\mathbf{0}$ is $\{\mathbf{a} + t\mathbf{d}: t\in\mathbb{R}\}$ (vector/parametric form). In $\mathbb{R}^3$ it can also be written $\frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3}$ (Cartesian form, when all $d_i\ne0$). The line through points $\mathbf{p}, \mathbf{q}$ has direction $\mathbf{q} - \mathbf{p}$.

The plane through $\mathbf{a}$ with normal $\mathbf{n}\ne\mathbf{0}$ is $\{\mathbf{x}: \mathbf{n}\cdot(\mathbf{x} - \mathbf{a}) = 0\}$, i.e. $n_1x + n_2y + n_3z = d$ with $d = \mathbf{n}\cdot\mathbf{a}$. Parametrically it is $\mathbf{a} + s\mathbf{u} + t\mathbf{v}$ for non-parallel $\mathbf{u}, \mathbf{v}$ in the plane, and then $\mathbf{n} = \mathbf{u}\times\mathbf{v}$.

The distance from $\mathbf{p}$ to the plane $\mathbf{n}\cdot\mathbf{x} = d$ is $\frac{|\mathbf{n}\cdot\mathbf{p} - d|}{\|\mathbf{n}\|}$. The distance from $\mathbf{p}$ to the line $\mathbf{a} + t\mathbf{d}$ in $\mathbb{R}^3$ is $\frac{\|(\mathbf{p} - \mathbf{a})\times\mathbf{d}\|}{\|\mathbf{d}\|}$.

• Line and plane: substitute $\mathbf{a} + t\mathbf{d}$ into $\mathbf{n}\cdot\mathbf{x} = d$ and solve for $t$ (no solution if $\mathbf{n}\cdot\mathbf{d} = 0$ and the line is off the plane).
• Two planes: if the normals aren't parallel they meet in a line with direction $\mathbf{n}_1\times\mathbf{n}_2$.
• Two lines in $\mathbb{R}^3$: equate $\mathbf{a} + t\mathbf{d} = \mathbf{b} + s\mathbf{e}$; they may meet, be parallel, or be skew (neither).

• Perpendicular? dot product $0$. Parallel? cross product $\mathbf{0}$ (in $\mathbb{R}^3$) or one a multiple of the other.
• Need a normal? cross two directions in the plane. Check it by dotting with both.
• Distances: project onto a normal (plane) or use the cross product (line).
• Sanity checks: substitute points back into equations; distances are non-negative; angles in $[0, \pi]$.