Planes

Consider the following vector equation for a plane in $\mathbb{R}^{4}$:

Find a new vector equation for the same plane by doing the change of parameters $t=1-r_{1}$, $s=r_{1}+r_{2}$.

The chapter mentions that the normal equation and standard equation of a plane only work in $\mathbb{R}^{3}$, and not in general $\mathbb{R}^{n}$. Why does the equation $ax+by+cz+dw=e$ not describe a plane in $\mathbb{R}^{4}$?

In Example 3.46, we calculated the angle $\theta$ between a line and a plane by calculating the angle $\phi$ between the direction vector of the line and the normal vector of the plane according to the formula

and then taking $\theta=\frac{\pi}{2}-\phi$.

1. Explain what happens when the dot product is negative. How should we adjust the formula to ensure that $\theta$ is always between $0$ and $\frac{\pi}{2}$?

2. Show that one can get the answer in a single s…

Use Exercise an exercise given elsewhere in the book to find an equation of a plane containing the two vectors $\mathbf{p}$ and $\mathbf{q}$ and the point $0$. Hint: If $(x,y,z)$ is a point in this plane, the volume of the parallelepiped determined by $(x,y,z)$ and the vectors $\mathbf{p}$, $\mathbf{q}$ equals 0.