Lines, Planes, and Geometric Transformations

Line in $\mathbb{R}^{3}$ — point $P_0$ and direction $\vec{d}$:

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

Plane in $\mathbb{R}^{3}$ — point $P_0$ and normal $\vec{n}=(a,b,c)$:

ax+by+cz = d

Key formulas

• Distance (point to plane):

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

• Angle between planes:

$$\cos\theta = \frac{|\vec{n}_1\cdot\vec{n}_2|}{\|\vec{n}_1\|\,\|\vec{n}_2\|}$$

• Parallel: $\vec{n}_1 \times \vec{n}_2 = \vec{0}$
• Line-plane intersection: substitute $\vec{r}(t)$ into the plane equation

Geometric transformations compose as matrix products. In homogeneous coordinates rotation + translation becomes a single matrix multiplication.

Tip: The intersection of two planes is a line with direction $\vec{n}_1\times\vec{n}_2$.

Common pitfall: A plane’s normal vector is perpendicular to every vector in the plane — the normal’s components come from the plane’s equation coefficients, not from points on the plane.