Line and Plane

Planes0 / 6

1. Equation of a plane from a normalThrough $\vec a$ with normal $\vec n$: $\vec r\cdot\vec n = \vec a\cdot\vec n$, i.e. $ax + by + cz = d$; normal form $\vec r\cdot\hat n = p$ with $p$ the distance from OStart
2. Plane through three points or containing linesTake the normal as a cross product of two directions in the plane, or set the determinant of $\vec r - \vec a$
3. Foot of the perpendicular and interceptsThe foot from O to $ax + by + cz = d$ is $\dfrac{d}{a^2 + b^2 + c^2}(a, b, c)$;
4. Distance of a point from a plane$d = \dfrac{|ax_1 + by_1 + cz_1 - d|}{\sqrt{a^2 + b^2 + c^2}}$,
5. Angle between planes, and a line and a planeBetween planes, the angle between their normals; between a line and a plane, $\sin\theta = \dfrac{|\vec b\cdot\vec n|}{|\vec b||\vec n|}$
6. A line meeting or lying in a planePut the line's general point into the plane to find where they meet; the line lies in the plane when $\vec b\cdot\vec n = 0$