Three Dimensional Geometry

Direction cosines and ratios

$l = cos α, m = cos β, n = cos γ;$ $l^{2} + m^{2} + n^{2} = 1$
From direction ratios
$l = \frac{a}{a ^{2} + b ^{2} + c ^{2}}, \dots$
Angle between lines
$cos θ = \frac{a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2}}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$

Equation
$r = a + λ b;$ $\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = \frac{z - z _{1}}{c}$
Shortest distance (skew lines)
$d = \frac{( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} )}{∣ b _{1} \times b _{2} ∣}$
Parallel lines
$d = \frac{∣ ( a _{2} - a _{1} ) \times b ∣}{∣ b ∣}$
Coplanar lines
$(a_{2} - a_{1}) \cdot (b_{1} \times b_{2}) = 0$

Equations
$a x + b y + cz + d = 0;$ $r \cdot \overset{n}{^} = p;$ $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
Through three points
$x - x_{1} x_{2} - x_{1} x_{3} - x_{1} y - y_{1} y_{2} - y_{1} y_{3} - y_{1} z - z_{1} z_{2} - z_{1} z_{3} - z_{1} = 0$
Point to plane
$d = \frac{∣ a x _{1} + b y _{1} + c z _{1} + d ∣}{a ^{2} + b ^{2} + c ^{2}}$
Parallel planes
$d = \frac{∣ d _{1} - d _{2} ∣}{a ^{2} + b ^{2} + c ^{2}}$
Family
$P_{1} + λ P_{2} = 0$ passes through the line $P_{1} = P_{2} = 0$

Angle
$sin θ = \frac{∣ b \cdot n ∣}{∣ b ∣∣ n ∣}$
Foot and image of a point
$\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = \frac{z - z _{1}}{c} = - \frac{a x _{1} + b y _{1} + c z _{1} + d}{a ^{2} + b ^{2} + c ^{2}}$ (foot) $;$ $\times 2$ for the image
Line lies in plane
$b \cdot n = 0$ and a point of the line satisfies the plane