Vector Formulas

$\vert\vert\textbf{v}\vert\vert = \sqrt{ v_x^2 + v_y^2 + v_z^2 + ...}$	Vector Magnitude
$\vert\vert\textbf{v}\vert\vert = \sqrt{ \textbf{v} \cdot \textbf{v} }$
$\vert\vert t\textbf{v}\vert\vert = \vert t\vert \vert\vert\textbf{v}\vert\vert$	Scalar magnitude property
$\textbf{a} \cdot \textbf{b} = a_x b_x + a_y b_y + a_x b_z + ...$	Dot product of vectors
$\textbf{a} \cdot \textbf{b} = \vert\vert\textbf{a}\vert\vert \vert\vert\textbf{b}\vert\vert cos\theta$
$\textbf{a} \cdot \textbf{a} = a^2$	Notation (indicates a scalar result)
a and b are orthogonal (perpindicular) if $\textbf{a} \cdot \textbf{b} = 0$
$\textbf{a} \times \textbf{b} = \begin{bmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end{bmatrix}$	Cross product on 3D vectors
$\textbf{a} \times \textbf{b} = \vert\vert\textbf{a}\vert\vert \vert\vert\textbf{b}\vert\vert cos \theta$
$A = \frac{1}{2} \vert\vert(\textbf{p}_1 - \textbf{p}_0) \times (\textbf{p}_2 - \textbf{p}_0)\vert\vert$	Area of a triangle defined by 3 points $\textbf{p}_0, \textbf{p}_1, \textbf{p}_2$
$\vert\vert\textbf{a} \times \textbf{b}\vert\vert = \vert\vert\textbf{a}\vert\vert \vert\vert\textbf{b}\vert\vert sin\theta$	Magnitude of a cross product
$\hat{\textbf{v}} = \frac{\textbf{v}}{\vert\vert\textbf{v}\vert\vert}$	Normalize a vector (becomes a unit vector, members sum to 1)
$\textbf{a} \times (\textbf{b} \times \textbf{c}) = \textbf{b}(\textbf{a} \cdot \textbf{c}) - \textbf{c}(\textbf{a} \cdot \textbf{b})$	Vector Triple Product (“back minus cab”)
$[ \textbf{a}, \textbf{b}, \textbf{c} ] = (\textbf{a} \times \textbf{b}) \cdot ... ...textbf{c}) \cdot \textbf{a} = (\textbf{c} \times \textbf{a}) \cdot \textbf{b}$	Scalar triple prodct. Equals the volume of a parallelepiped formed by a,b, and c.
$[ \textbf{a}, \textbf{b}, \textbf{c} ] = -[ \textbf{c}, \textbf{b}, \textbf{a} ]$
$\textbf{a}_{\vert\vert\textbf{b}} = \frac{\textbf{a} \cdot \textbf{b}}{b^2} \textbf{b}$	Projection of a onto b (cosine/X component)
$\textbf{a}_{\vert\vert\textbf{b}} = \frac{1}{b^2} \begin{bmatrix} b_x^2 & b_x ... ... b_z^2 \\ \end{bmatrix} \begin{bmatrix} a_x \\ a_y \\ a_z \\ \end{bmatrix}$	Projection of a onto b with 3d vectors
$\textbf{a}_{\perp \textbf{b}} = \textbf{a} - \textbf{a}_{\vert\vert\textbf{b}}$	Rejection of b from a (sin/Y component)
$(\textbf{a}_{\vert\vert\textbf{b}})^2 + (\textbf{a}_{\perp \textbf{b}})^2 = a^2$	Projection/Rejection property (pythagorean theorem)
$\textbf{u}_1 = \textbf{v}_1, \textbf{u}_2 = \textbf{v}_2 - (\textbf{v}_2)_{\ve... ...{v}_3)_{\vert\vert\textbf{u}_1} - (\textbf{v}_3)_{\vert\vert\textbf{u}_2}, ...$	Gram-Schmidt process (orthogonal vectors set).
	Common to normalize each result (orthonormalization)
$\textbf{a} \otimes \textbf{b} = \textbf{a} \textbf{b}^T = \begin{bmatrix}a_x \\ a_y \\ a_z \\ \end{bmatrix}\begin{bmatrix}b_x & b_y & b_z \\ \end{bmatrix}$	3D Vector Outer Product
$= \begin{bmatrix} a_x b_z & a_x b_y & a_x b_z \\ a_y b_x & a_y b_y & a_y b_z \\ a_z b_x & a_z b_y & a_z b_z \\ \end{bmatrix}$
$\textbf{v} = (v_x, v_y, v_z, 0)$	Homogenous direction vector
$\textbf{\textit{p}} = (v_x, v_y, v_z, 1)$	Homogenous position vector