Problem 23
Let \(A\) and \(B\) be points other than the origin, and let \(\mathbf{a}\) and \(\mathbf{b}\) be their vectors. If \(\mathbf{a}\) and \(\mathbf{b}\) are not parallel, show that the plane through \(A, B\), and the origin is given by $$ \left\\{P(x, y, z) \mid\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=s \mathbf{a}+t \mathbf{b} \text { for some } s \text { and } t\right\\} $$
Problem 23
Find the shortest distance between the following pairs of parallel lines. $$ \begin{array}{l} \text { a. }\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} 2 \\ -1 \\ 3 \end{array}\right]+t\left[\begin{array}{r} 1 \\ -1 \\ 4 \end{array}\right] ; \\ {\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]+t\left[\begin{array}{r} 1 \\ -1 \\ 4 \end{array}\right]} \\ \text { b. }\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 3 \\ 0 \\ 2 \end{array}\right]+t\left[\begin{array}{l} 3 \\ 1 \\ 0 \end{array}\right] ; \\ \quad\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} -1 \\ 2 \\ 2 \end{array}\right]+t\left[\begin{array}{l} 3 \\ 1 \\ 0 \end{array}\right] \end{array} $$
Problem 23
In each case, verify that the points \(P\) and \(Q\) lie on the line. $$ \begin{array}{ll} \text { a. } & x=3-4 t \quad P(-1,3,0), Q(11,0,3) \\ & y=2+t \\ & z=1-t \\ \text { b. } & x=4-t \quad P(2,3,-3), Q(-1,3,-9) \\ & y=3 \\ & z=1-2 t \end{array} $$
Problem 24
Find the shortest distance between the following pairs of nonparallel lines and find the points on the lines that are closest together. a. \(\left[\begin{array}{l}x \\ y \\\ z\end{array}\right]=\left[\begin{array}{l}3 \\ 0 \\\ 1\end{array}\right]+s\left[\begin{array}{r}2 \\ 1 \\ -3\end{array}\right] ;\) \(\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{r}1 \\\ 1 \\ -1\end{array}\right]+t\left[\begin{array}{l}1 \\ 0 \\\ 1\end{array}\right]\) b. \(\left[\begin{array}{l}x \\ y \\\ z\end{array}\right]=\left[\begin{array}{r}1 \\ -1 \\\ 0\end{array}\right]+s\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right] ;\) \(\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{r}2 \\\ -1 \\ 3\end{array}\right]+t\left[\begin{array}{l}3 \\ 1 \\\ 0\end{array}\right]\) c. \(\left[\begin{array}{l}x \\ y \\\ z\end{array}\right]=\left[\begin{array}{r}3 \\ 1 \\\ -1\end{array}\right]+s\left[\begin{array}{r}1 \\ 1 \\ -1\end{array}\right]\); \(\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\\ 2 \\ 0\end{array}\right]+t\left[\begin{array}{l}1 \\ 0 \\\ 2\end{array}\right]\) d. \(\left[\begin{array}{l}x \\ y \\\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 2 \\\ 3\end{array}\right]+s\left[\begin{array}{r}2 \\ 0 \\ -1\end{array}\right] ;\) \(\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{r}3 \\\ -1 \\ 0\end{array}\right]+t\left[\begin{array}{l}1 \\ 1 \\\ 0\end{array}\right]\)
Problem 24
Find the point of intersection (if any) of the following pairs of lines. $$ \text { a. } \begin{array}{ll} x=3+t & x=4+2 s \\ & y=1-2 t & y=6+3 s \\ & z=3+3 t & z=1+s \\ & x=1-t & x=2 s \end{array} $$ b. \(\quad y=2+2 t \quad y=1+s\) $$ \begin{array}{c} z=-1+3 t \quad z=3 \\ \text { c. }\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} 3 \\ -1 \\ 2 \end{array}\right]+t\left[\begin{array}{r} 1 \\ 1 \\ -1 \end{array}\right] \\ {\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} 1 \\ 1 \\ -2 \end{array}\right]+s\left[\begin{array}{l} 2 \\ 0 \\ 3 \end{array}\right]} \end{array} $$ d. \(\left[\begin{array}{l}x \\ y \\\ z\end{array}\right]=\left[\begin{array}{r}4 \\ -1 \\\ 5\end{array}\right]+t\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]\) \(\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{r}2 \\\ -7 \\ 12\end{array}\right]+s\left[\begin{array}{r}0 \\ -2 \\\ 3\end{array}\right]\)
Problem 24
Let \(A\) be a \(2 \times 3\) matrix of rank 2 with rows \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\). Show that $$P=\\{X A \mid X=[x y] ; x, y \text { arbitrary }\\}$$ is the plane through the origin with normal \(\mathbf{r}_{1} \times \mathbf{r}_{2}\).
Problem 25
Show that if a line passes through the origin, the vectors of points on the line are all scalar multiples of some fixed nonzero vector.
Problem 25
Given the cube with vertices \(P(x, y, z)\) where each of \(x, y,\) and \(z\) is either 0 or \(2,\) consider the plane perpendicular to the diagonal through \(P(0,0,0)\) and \(P(2,2,2)\) and bisecting it. a. Show that the plane meets six of the edges of the cube and bisects them. b. Show that the six points in (a) are the vertices of a regular hexagon.
Problem 25
Show that two lines in the plane with slopes \(m_{1}\) and \(m_{2}\) are perpendicular if and only if \(m_{1} m_{2}=-1 .[\) Hint : Example \(4.1 .11 .]\)
Problem 26
a. Show that, of the four diagonals of a cube, no pair is perpendicular. b. Show that each diagonal is perpendicular to the face diagonals it does not meet.
