Quick questions on Planes in 3D explained: H2 Mathematics Vectors and Complex Numbers

A plane with normal vector $\mathbf{n}$ passing through a point with position vector $\mathbf{a}$ satisfies

If two vectors lie in the plane, their cross product is a normal. This is how you build a plane through three given points: take two edge vectors and cross them.

Substitute the parametric point of the line into the plane equation, solve for the parameter $\lambda$, and back-substitute to get the point. If the line is parallel to the plane (its direction is perpendicular to $\mathbf{n}$), there is either no solution or the line lies in the plane.

Two non-parallel planes intersect in a line. Solve their Cartesian equations simultaneously (setting one variable as a parameter) to get the line, or take the cross product of the two normals to find the direction of the line of intersection.

If the direction is perpendicular to $\mathbf{n}$, check whether the line lies in the plane (any point satisfies it) or misses it entirely.

Take the absolute value; distance is non-negative.

State the normal vector of the plane $3x - y + 4z = 2$. [1 mark]

Find where the line $\mathbf{r} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} + \lambda\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ meets the plane $x + y + z = 6$. [3 marks]

Explain how to find the angle between two planes. [2 marks]