Vector Formulas and Rules

Scalar (or Dot) Product

$\vec a$ and $\vec b$ are vectors given by their components as follows
$\vec a = \lt a_1, a_2 , a_3 \gt$ and $\vec b = \lt b_1, b_2 , b_3 \gt$
The scalar (or dot) product of vectors $\vec a$ and $\vec b$ is defined by
$\vec a \cdot \vec b = || \vec a || \; || \vec b || \cos \theta = \sum_{i=1}^{3} a_i b_i = a_1 b_1 + a_2 b_2 + a_3 b_3$
where $\theta$ is the angle between vectors $\vec a$ and $\vec b$ and $|| \vec a ||$ and $|| \vec b ||$ are their magnitudes.

Vector (or Cross) Product

The vector (or cross ) product of vectors $\vec a$ and $\vec b$ is defined by
$\vec a \times \vec b = || \vec a || \; || \vec b || \sin \theta \; \vec n = \begin{vmatrix} \vec i & \vec j & \vec k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{vmatrix}$

$\quad \quad = \vec i \begin{vmatrix} a_2 & a_3 \\ b_2 & b_3 \\ \end{vmatrix} - \vec j \begin{vmatrix} a_1 & a_3 \\ b_1 & b_3 \\ \end{vmatrix} + \vec k \begin{vmatrix} a_1 & a_2 \\ b_1 & b_2 \\ \end{vmatrix}$

$\quad \quad = (a_2 b_3 - a_3 b_2) \vec i - (a_1 b_3 - a_3 b_1) \vec j + (a_1b_2 - a_2 b_1) \vec k$
$\vec n$ is a unit vector perpendicular to vectors $\vec a$ and $\vec b$, such that $\vec a$, $\vec b$ and $\vec n$ form a right-handed set of vectors.

Scalar Triple Product

The scalar triple product of $\vec a$, $\vec b$ and $\vec c$ is given by
$(\vec a \times \vec b) \cdot \vec c = \vec a \cdot (\vec b \times \vec c)$
$\quad \quad = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{vmatrix}$

$= a_1 \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \\ \end{vmatrix} - a_2 \begin{vmatrix} b_1 & b_3 \\ c_1 & c_3 \\ \end{vmatrix} + a_3 \begin{vmatrix} b_1 & b_2 \\ c_1 & c_2 \\ \end{vmatrix}$
$= a_1( b_2 c_3 - b_3 c_2) - a_2 (b_1 c_3 -b_3 c_1) + a_3 (b_1 c_2 - b_2 c_1)$

Vector (Cross) Triple Product

$\vec a \times (\vec b \times \vec c) = (\vec a \cdot \vec c) \vec b - (\vec a \cdot \vec b) \vec c$