Vector Lines and Planes

$\mathbf{r} = \mathbf{a} + t\mathbf{d}$

where $\mathbf{a}$ is a point on the line and $\mathbf{d}$ is the direction vector.

$\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}$

where $\mathbf{n}$ is the normal vector.

Cartesian form: $ax + by + cz = d$ where $\mathbf{n} = \binom{a}{b}{c}$.

Angle Between Lines

$\cos\theta = \frac{|\mathbf{d_1} \cdot \mathbf{d_2}|}{|\mathbf{d_1}||\mathbf{d_2}|}$

Angle Between Line and Plane

$\sin\theta = \frac{|\mathbf{d} \cdot \mathbf{n}|}{|\mathbf{d}||\mathbf{n}|}$

Intersection of Line and Plane

Substitute parametric equations into the plane equation.