Problem 8
Find the three internal angles of the triangle with vertices: a. \(A(3,1,-2), B(3,0,-1),\) and \(C(5,2,-1)\) b. \(A(3,1,-2), B(5,2,-1),\) and \(C(4,3,-3)\)
Problem 9
Show that the line through \(P_{0}(3,1,4)\) and \(P_{1}(2,1,3)\) is perpendicular to the line through \(P_{2}(1,-1,2)\) and \(P_{3}(0,5,3)\)
Problem 9
Let \(\mathbf{u}\) and \(\mathbf{v}\) be nonzero, nonorthogonal vectors. If \(\theta\) is the angle between them, show that \(\tan \theta=\frac{\|\mathbf{u} \times \mathbf{v}\|}{\mathbf{u} \cdot \mathbf{v}}\).
Problem 9
Let \(L\) be the line through the origin in \(\mathbb{R}^{2}\) with direction vector \(\mathbf{d}=\left[\begin{array}{l}a \\ b\end{array}\right] \neq 0\) a. If \(P_{L}\) denotes projection on \(L\), show that \(P_{L}\) has matrix \(\frac{1}{a^{2}+b^{2}}\left[\begin{array}{cc}a^{2} & a b \\ a b & b^{2}\end{array}\right]\) b. If \(Q_{L}\) denotes reflection in \(L,\) show that \(Q_{L}\) has ma\(\operatorname{trix} \frac{1}{a^{2}+b^{2}}\left[\begin{array}{cc}a^{2}-b^{2} & 2 a b \\ 2 a b & b^{2}-a^{2}\end{array}\right]\)
Problem 9
In each case, find \(\overrightarrow{P Q}\) and \(\|\overrightarrow{P Q}\| .\) a. \(P(1,-1,3), Q(3,1,0)\) b. \(P(2,0,1), Q(1,-1,6)\) c. \(P(1,0,1), Q(1,0,-3)\) d. \(P(1,-1,2), Q(1,-1,2)\) e. \(P(1,0,-3), Q(-1,0,3)\) f. \(P(3,-1,6), Q(1,1,4)\)
Problem 10
Show that points \(A, B,\) and \(C\) are all on one line if and only if \(\overrightarrow{A B} \times \overrightarrow{A C}=0\).
Problem 10
In each case, compute the projection of \(\mathbf{u}\) on \(\mathbf{v}\) a. \(\mathbf{u}=\left[\begin{array}{l}5 \\ 7 \\ 1\end{array}\right], \mathbf{v}=\left[\begin{array}{r}2 \\ -1 \\ 3\end{array}\right]\) b. \(\mathbf{u}=\left[\begin{array}{r}3 \\ -2 \\ 1\end{array}\right], \mathbf{v}=\left[\begin{array}{l}4 \\ 1 \\ 1\end{array}\right]\) c. \(\mathbf{u}=\left[\begin{array}{r}1 \\ -1 \\ 2\end{array}\right], \mathbf{v}=\left[\begin{array}{r}3 \\ -1 \\ 1\end{array}\right]\) d. \(\mathbf{u}=\left[\begin{array}{r}3 \\ -2 \\ -1\end{array}\right], \mathbf{v}=\left[\begin{array}{r}-6 \\ 4 \\ 2\end{array}\right]\)
Problem 10
In each case, find a point \(Q\) such that \(\overrightarrow{P Q}\) has (i) the same direction as \(\mathbf{v}\); (ii) the opposite direction to \(\mathbf{v}\). a. \(P(-1,2,2), \mathbf{v}=\left[\begin{array}{l}1 \\ 3 \\\ 1\end{array}\right]\) b. \(P(3,0,-1), \mathbf{v}=\left[\begin{array}{r}2 \\ -1 \\\ 3\end{array}\right]\)
Problem 11
Show that points \(A, B,\) and \(C\) are all on one line if and only if \(\overrightarrow{A B} \times \overrightarrow{A C}=0\).
Problem 11
In each case, write \(\mathbf{u}=\mathbf{u}_{1}+\mathbf{u}_{2},\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v}\). a. \(\mathbf{u}=\left[\begin{array}{r}2 \\ -1 \\ 1\end{array}\right], \mathbf{v}=\left[\begin{array}{r}1 \\ -1 \\ 3\end{array}\right]\) b. \(\mathbf{u}=\left[\begin{array}{l}3 \\ 1 \\ 0\end{array}\right], \mathbf{v}=\left[\begin{array}{r}-2 \\ 1 \\ 4\end{array}\right]\) c. \(\mathbf{u}=\left[\begin{array}{r}2 \\ -1 \\ 0\end{array}\right], \mathbf{v}=\left[\begin{array}{r}3 \\ 1 \\ -1\end{array}\right]\) d. \(\mathbf{u}=\left[\begin{array}{r}3 \\ -2 \\ 1\end{array}\right], \mathbf{v}=\left[\begin{array}{r}-6 \\ 4 \\ -1\end{array}\right]\)
