Chapter 4

Show that points \(A, B,\) and \(C\) are all on one line if and only if \(\overrightarrow{A B} \times \overrightarrow{A C}=0\).

Let \(\mathbf{u}=\left[\begin{array}{r}3 \\ -1 \\ 0\end{array}\right], \mathbf{v}=\left[\begin{array}{l}4 \\ 0 \\ 1\end{array}\right],\) and \(\mathbf{w}=\left[\begin{array}{c}-1 \\ 1 \\ 5\end{array}\right] .\) In each case, find \(\mathbf{x}\) such that: a. \(3(2 \mathbf{u}+\mathbf{x})+\mathbf{w}=2 \mathbf{x}-\mathbf{v}\) b. \(2(3 \mathbf{v}-\mathbf{x})=5 \mathbf{w}+\mathbf{u}-3 \mathbf{x}\)

Let \(\mathbf{u}=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right], \mathbf{v}=\left[\begin{array}{l}0 \\ 1 \\ 2\end{array}\right],\) and \(\mathbf{w}=\left[\begin{array}{r}1 \\ 0 \\ -1\end{array}\right] .\) In each case, find numbers \(a, b,\) and \(c\) such that \(\mathbf{x}=a \mathbf{u}+b \mathbf{v}+c \mathbf{w}\) a. \(\mathbf{x}=\left[\begin{array}{r}2 \\ -1 \\ 6\end{array}\right]\) b. \(\mathbf{x}=\left[\begin{array}{l}1 \\ 3 \\ 0\end{array}\right]\)

Let \(\mathbf{u}=\left[\begin{array}{r}3 \\ -1 \\ 0\end{array}\right], \mathbf{v}=\left[\begin{array}{l}4 \\ 0 \\ 1\end{array}\right],\) and \(\mathbf{z}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right] .\) In each case, show that there are no numbers \(a, b,\) and \(c\) such that: a. \(a \mathbf{u}+b \mathbf{v}+c \mathbf{z}=\left[\begin{array}{l}1 \\ 2 \\\ 1\end{array}\right]\) b. \(a \mathbf{u}+b \mathbf{v}+c \mathbf{z}=\left[\begin{array}{r}5 \\ 6 \\\ -1\end{array}\right]\)

Compute \(\mathbf{u} \times \mathbf{v}\) where: a. \(\mathbf{u}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right], \mathbf{v}=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]\) b. \(\mathbf{u}=\left[\begin{array}{r}3 \\ -1 \\ 0\end{array}\right], \mathbf{v}=\left[\begin{array}{r}-6 \\ 2 \\ 0\end{array}\right]\) c. \(\mathbf{u}=\left[\begin{array}{r}3 \\ -2 \\ 1\end{array}\right], \mathbf{v}=\left[\begin{array}{r}1 \\ 1 \\ -1\end{array}\right]\) d. \(\mathbf{u}=\left[\begin{array}{r}2 \\ 0 \\ -1\end{array}\right], \mathbf{v}=\left[\begin{array}{l}1 \\ 4 \\ 7\end{array}\right]\)

Show that the volume of the parallelepiped determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u} \times \mathbf{v}\) is \(\|\mathbf{u} \times \mathbf{v}\|^{2}\).

Find an equation of each of the following planes. a. Passing through \(A(2,1,3), B(3,-1,5),\) and \(C(1,2,-3)\) b. Passing through \(A(1,-1,6), B(0,0,1),\) and \(C(4,7,-11)\) c. Passing through \(P(2,-3,5)\) and parallel to the plane with equation \(3 x-2 y-z=0\). d. Passing through \(P(3,0,-1)\) and parallel to the plane with equation \(2 x-y+z=3\). e. Containing \(P(3,0,-1)\) and the line \(\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}0 \\\ 0 \\ 2\end{array}\right]+t\left[\begin{array}{l}1 \\ 0 \\\ 1\end{array}\right]\) f. Containing \(P(2,1,0)\) and the line \(\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{r}3 \\\ -1 \\ 2\end{array}\right]+t\left[\begin{array}{r}1 \\ 0 \\\ -1\end{array}\right]\) g. \(\mathrm{e}^{-}\) \(\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{r}1 \\\ -1 \\ 2\end{array}\right]+t\left[\begin{array}{l}1 \\ 1 \\\ 1\end{array}\right]\) and \(\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}0 \\\ 0 \\ 2\end{array}\right]+t\left[\begin{array}{r}1 \\ -1 \\\ 0\end{array}\right]\) h. Containing the lines \(\left[\begin{array}{l}x \\ y \\\ z\end{array}\right]=\left[\begin{array}{l}3 \\ 1 \\\ 0\end{array}\right]+t\left[\begin{array}{r}1 \\ -1 \\ 3\end{array}\right]\) and \(\left[\begin{array}{l}x \\ y \\\ z\end{array}\right]=\left[\begin{array}{r}0 \\ -2 \\\ 5\end{array}\right]+t\left[\begin{array}{r}2 \\ 1 \\ -1\end{array}\right]\). i. Each point of which is equidistant from \(P(2,-1,3)\) and \(Q(1,1,-1)\) j. Each point of which is equidistant from \(P(0,1,-1)\) and \(Q(2,-1,-3)\)

Given \(P_{1}(2,1,-2)\) and \(P_{2}(1,-2,0)\) Find the coordinates of the point \(P\) : a. \(\frac{1}{5}\) the way from \(P_{1}\) to \(P_{2}\) b. \(\frac{1}{4}\) the way from \(P_{2}\) to \(P_{1}\)

In each case, find a vector equation of the line. a. Passing through \(P(3,-1,4)\) and perpendicular to the plane \(3 x-2 y-z=0\) b. Passing through \(P(2,-1,3)\) and perpendicular to the plane \(2 x+y=1\) c. Passing through \(P(0,0,0)\) and perpendicular $$ \begin{array}{l} \text { to the lines }\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right]+t\left[\begin{array}{r} 2 \\ 0 \\ -1 \end{array}\right] \text { and } \\ {\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} 2 \\ 1 \\ -3 \end{array}\right]+t\left[\begin{array}{r} 1 \\ -1 \\ 5 \end{array}\right]} \end{array} $$ d. Passing through \(P(1,1,-1)\), and perpendicular to the lines $$ \begin{array}{l} {\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 2 \\ 0 \\ 1 \end{array}\right]+t\left[\begin{array}{r} 1 \\ 1 \\ -2 \end{array}\right] \text { an }} \\ {\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} 5 \\ 5 \\ -2 \end{array}\right]+t\left[\begin{array}{r} 1 \\ 2 \\ -3 \end{array}\right]} \end{array} $$ e. Passing through \(P(2,1,-1)\), intersecting the line \(\left[\begin{array}{l}x \\\ y \\ z\end{array}\right]=\left[\begin{array}{r}1 \\ 2 \\\ -1\end{array}\right]+t\left[\begin{array}{l}3 \\ 0 \\ 1\end{array}\right],\) and perpendicular to that line. f. Passing through \(P(1,1,2)\), intersecting the line \(\left[\begin{array}{l}x \\\ y \\ z\end{array}\right]=\left[\begin{array}{l}2 \\ 1 \\\ 0\end{array}\right]+t\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right],\) and perpendicular to line.

Find the two points trisecting the segment between \(P(2,3,5)\) and \(Q(8,-6,2)\).