Mathematics · Ch 5 — Introduction to Three-Dimensional Geometry

Summary

The coordinate axes ( $x$ , $y$ , $z$ ) are mutually perpendicular lines through the origin $O(0,0,0)$ . The coordinate planes are $xy$ -plane ( $z=0$ ), $yz$ -plane ( $x=0$ ), and $zx$ -plane ( $y=0$ ). Any point in space is written as an ordered triple $(x, y, z)$ .
Distance formula between $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ :

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

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

For external division, replace $n$ with $-n$ .

Midpoint of $PQ$ is $\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right)$ .
Coordinates of centroid of a triangle with vertices $(x_1,y_1,z_1)$ , $(x_2,y_2,z_2)$ , $(x_3,y_3,z_3)$ :

$\left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3} \right)$ …