Key Equations

Distance between two points in space:	$d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}$
Sphere with center $(a, b, c)$ and radius r:	${(x - a)}^{2} + {(y - b)}^{2} + {(z - c)}^{2} = r^{2}$

Dot product of u and v	$\begin{matrix} u \cdot v & = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3} \\ = ‖ u ‖ ‖ v ‖ cos θ \end{matrix}$
Cosine of the angle formed by $u$ and $v$	$cos θ = \frac{u \cdot v}{‖ u ‖ ‖ v ‖}$
Vector projection of $v$ onto $u$	${proj}_{u} v = \frac{u \cdot v}{{‖ u ‖}^{2}} u$
Scalar projection of $v$ onto $u$	${comp}_{u} v = \frac{u \cdot v}{‖ u ‖}$
Work done by a force F to move an object through displacement vector $\vec{P Q}$	$W = F \cdot \vec{P Q} = ‖ F ‖ ‖ \vec{P Q} ‖ cos θ$

The cross product of two vectors in terms of the unit vectors

u \times v = (u_{2} v_{3} - u_{3} v_{2}) i - (u_{1} v_{3} - u_{3} v_{1}) j + (u_{1} v_{2} - u_{2} v_{1}) k

Vector Equation of a Line	$r = r_{0} + t v$
Parametric Equations of a Line	$\frac{x - x_{0}}{a} = \frac{y - y_{0}}{b} = \frac{z - z_{0}}{c}$
Vector Equation of a Plane	$n \cdot \vec{P Q} = 0$
Scalar Equation of a Plane	$a (x - x_{0}) + b (y - y_{0}) + c (z - z_{0}) = 0$
Distance between a Plane and a Point	$d = ‖ {proj}_{n} \vec{Q P} ‖ = \| {comp}_{n} \vec{Q P} \| = \frac{\| \vec{Q P} \cdot n \|}{‖ n ‖}$