Introduction To Three- Dimensional Geometry — Formula Sheet

PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

OP = \sqrt{x^2 + y^2 + z^2}

\left( \frac{l x_2 + m x_1}{l + m}, \frac{l y_2 + m y_1}{l + m}, \frac{l z_2 + m z_1}{l + m} \right)

\left( \frac{l x_2 - m x_1}{l - m}, \frac{l y_2 - m y_1}{l - m}, \frac{l z_2 - m z_1}{l - m} \right)

\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)

l = \cos \alpha, m = \cos \beta, n = \cos \gamma

l^2 + m^2 + n^2 = 1

l = \pm \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \pm \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \pm \frac{c}{\sqrt{a^2 + b^2 + c^2}}

Distance from origin to point P(x, y, z) is √(x² + y² + z²).

l = cosα, m = cosβ, n = cosγ.

For any line

l² + m² + n² = 1.

Distance from origin is √(x² + y² + z²).

The distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is given by the formula: √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].