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 ‖}