Lines, Planes, and Geometric Transformations

Lines and Planes in Space

Geometry in 3D space relies on fundamental equations for lines and planes. A Line in 3D is defined by a starting point $P_0$ and direction vector $\vec{d}$ using parameter $t$:

$$\vec{r}(t) = P_0 + t\,\vec{d}$$

A Plane in 3D uses a point $P_0$ and a normal vector $\vec{n}=(a,b,c)$ where $ax+by+cz = d$.

Common Pitfall: A plane's normal vector is perpendicular to every vector in the plane. Its components come from the equation coefficients ($a,b,c$), never from points on the plane itself.

Distances and Transformations

Measure spatial relationships using core formulas. The Distance from point $(x_0,y_0,z_0)$ to a plane is:

$$D = \frac{|ax_0+by_0+cz_0-d|}{\sqrt{a^2+b^2+c^2}}$$

Geometric transformations compose as matrix products. Using homogeneous coordinates, rotation and translation combine into a single matrix multiplication.