Chapter 4

Find the equations of the line of intersection of the following planes. a. $2x-3y+2z=5$ and $x+2y-z=4$. b. $3x+y-2z=1$ and $x+y+z=5$.

Find all vectors $\mathbf{u}$ that are parallel to $\mathbf{v}=\left[\begin{array}{r}3\\ -2\\ 1\end{array}\right]$ and satisfy $\Vert \mathbf{u}\Vert =3\Vert \mathbf{v}\Vert .$

In each case, find all points of intersection of the given plane and the line $\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{r}1\\ -2\\ 3\end{array}\right]+t\left[\begin{array}{r}2\\ 5\\ -1\end{array}\right]$ a. $x-3y+2z=4$ b. $2x-y-z=5$ c. $3x-y+z=8$ d. $-x-4y-3z=6$

Let $P,Q,R,$ and $S$ be four points, $not$ all on one plane, as in the diagram. Show that the volume of the pyramid they determine is $$\frac{1}{6}|\overrightarrow{PQ}\cdot (\overrightarrow{PR}\times \overrightarrow{PS})|$$

Let $P,Q,$ and $R$ be the vertices of a parallelogram with adjacent sides $PQ$ and $PR$. In each case, find the other vertex $S$. $$\text{ a. }P(3,-1,-1),Q(1,-2,0),R(1,-1,2)$$ b. $P(2,0,-1),Q(-2,4,1),R(3,-1,0)$

Find the equation of $all$ planes: a. Perpendicular to the line $\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{r}2\\ -1\\ 3\end{array}\right]+t\left[\begin{array}{l}2\\ 1\\ 3\end{array}\right]$ b. Perpendicular to the line $\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{r}1\\ 0\\ -1\end{array}\right]+t\left[\begin{array}{l}3\\ 0\\ 2\end{array}\right]$ c. Containing the origin. d. Containing $P(3,2,-4)$. e. Containing $P(1,1,-1)$ and $Q(0,1,1)$. f. Containing $P(2,-1,1)$ and $Q(1,0,0)$. g. Containing the line $\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}2\\ 1\\ 0\end{array}\right]+t\left[\begin{array}{r}1\\ -1\\ 0\end{array}\right]$ h. Containing the line $\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}3\\ 0\\ 2\end{array}\right]+t\left[\begin{array}{r}1\\ -2\\ -1\end{array}\right]$

In each case either prove the statement or give an example showing that it is false. a. The zero vector $\mathbf{0}$ is the only vector of length 0 . b. If $\Vert \mathbf{v}-\mathbf{w}\Vert =0,$ then $\mathbf{v}=\mathbf{w}$. c. If $\mathbf{v}=-\mathbf{v},$ then $\mathbf{v}=\mathbf{0}$. d. If $\Vert \mathbf{v}\Vert =\Vert \mathbf{w}\Vert ,$ then $\mathbf{v}=\mathbf{w}$. e. If $\Vert \mathbf{v}\Vert =\Vert \mathbf{w}\Vert ,$ then $\mathbf{v}=\pm \mathbf{w}$. f. If $\mathbf{v}=t\mathbf{w}$ for some scalar $t,$ then $\mathbf{v}$ and $\mathbf{w}$ have the same direction. $\mathrm{g}$. If $\mathbf{v},\mathbf{w},$ and $\mathbf{v}+\mathbf{w}$ are nonzero, and $\mathbf{v}$ and $\mathbf{v}+\mathbf{w}$ parallel, then $\mathbf{v}$ and $\mathbf{w}$ are parallel. h. $\Vert -5\mathbf{v}\Vert =-5\Vert \mathbf{v}\Vert ,$ for all $\mathbf{v}$. i. If $\Vert \mathbf{v}\Vert =\Vert 2\mathbf{v}\Vert ,$ then $\mathbf{v}=\mathbf{0}$. j. $\Vert \mathbf{v}+\mathbf{w}\Vert =\Vert \mathbf{v}\Vert +\Vert \mathbf{w}\Vert ,$ for all $\mathbf{v}$ and $\mathbf{w}$.

Find the vector and parametric equations of the following lines. a. The line parallel to $\left[\begin{array}{r}2\\ -1\\ 0\end{array}\right]$ and passing through $P(1,-1,3)$ b. The line passing through $P(3,-1,4)$ and $Q(1,0,-1)$ c. The line passing through $P(3,-1,4)$ and $Q(3,-1,5)$ d. The line parallel to $\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]$ and passing through $P(1,1,1)$ e. The line passing through $P(1,0,-3)$ and parallel to the line with parametric equations $x=-1+2t$, $y=2-t,$ and $z=3+3t$ f. The line passing through $P(2,-1,1)$ and parallel to the line with parametric equations $x=2-t,$ $y=1,$ and $z=t$ $\mathrm{g}$. The lines through $P(1,0,1)$ that meet the line with vector equation $\mathbf{p}=\left[\begin{array}{l}1\\ 2\\ 0\end{array}\right]+t\left[\begin{array}{r}2\\ -1\\ 2\end{array}\right]$ at points at distance 3 from $P_0(1,2,0)$.

If a plane contains two distinct points $P_1$ and $P_2$, show that it contains every point on the line through $P_1$ and $P_2$.

Show that the (shortest) distance between two planes $\mathbf{n}\cdot \mathbf{p}=d_1$ and $\mathbf{n}\cdot \mathbf{p}=d_2$ with $\mathbf{n}$ as normal is $\frac{|d_2-d_1|}{\Vert \mathbf{n}\Vert }$.