Problem 6
Consider the triangle with vertices \(P(2,0,-3), Q(5,-2,1),\) and \(R(7,5,3)\) a. Show that it is a right-angled triangle. b. Find the lengths of the three sides and verify the Pythagorean theorem.
Problem 6
Let \(P_{0}\) be a point with vector \(\mathbf{p}_{0},\) and let \(a x+b y+c z=d\) be the equation of a plane with normal \(\mathbf{n}=\left[\begin{array}{l}a \\\ b \\ c\end{array}\right]\) a. Show that the point on the plane closest to \(P_{0}\) has vector \(\mathbf{p}\) given by $$ \mathbf{p}=\mathbf{p}_{0}+\frac{d-\left(\mathbf{p}_{0} \cdot \mathbf{n}\right)}{\|\mathbf{n}\|^{2}} \mathbf{n} $$ \(\left[\right.\) Hint \(: \mathbf{p}=\mathbf{p}_{0}+t \mathbf{n}\) for some \(t,\) and \(\left.\mathbf{p} \cdot \mathbf{n}=\mathbf{d} .\right]\) b. Show that the shortest distance from \(P_{0}\) to the plane is \(\frac{\left|d-\left(\mathbf{p}_{0} \cdot \mathbf{n}\right)\right|}{\|\mathbf{n}\|}\). c. Let \(P_{0}^{\prime}\) denote the reflection of \(P_{0}\) in the planethat is, the point on the opposite side of the plane such that the line through \(P_{0}\) and \(P_{0}^{\prime}\) is perpendicular to the plane. Show that \(\mathbf{p}_{0}+2 \frac{d-\left(\mathbf{p}_{0} \cdot \mathbf{n}\right)}{\|\mathbf{n}\|^{2}} \mathbf{n}\) is the vector of \(P_{0}^{\prime}\)
Problem 6
Let \(A, B,\) and \(C\) denote the three vertice of a triangle. a. If \(E\) is the midpoint of side \(B C\), show that $$ \overrightarrow{A E}=\frac{1}{2}(\overrightarrow{A B}+\overrightarrow{A C}) $$ b. If \(F\) is the midpoint of side \(A C\), show that $$ \overrightarrow{F E}=\frac{1}{2} \overrightarrow{A B} $$
Problem 6
Find the matrix of the rotation about the \(y\) axis through the angle \(\theta\) (from the positive \(x\) axis to the positive \(z\) axis).
Problem 7
If \(A\) is \(3 \times 3,\) show that the image of the line in \(\mathbb{R}^{3}\) through \(\mathbf{p}_{0}\) with direction vector \(\mathbf{d}\) is the line through \(A \mathbf{p}_{0}\) with direction vector \(A\) d, assuming that \(A \mathbf{d} \neq \mathbf{0} .\) What happens if \(A \mathbf{d}=\mathbf{0} ?\)
Problem 7
Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are parallel in each of the following cases. a. \(\mathbf{u}=\left[\begin{array}{r}-3 \\ -6 \\ 3\end{array}\right] ; \mathbf{v}=\left[\begin{array}{r}5 \\ 10 \\ -5\end{array}\right]\) b. \(\mathbf{u}=\left[\begin{array}{r}3 \\ -6 \\ 3\end{array}\right] ; \mathbf{v}=\left[\begin{array}{r}-1 \\ 2 \\ -1\end{array}\right]\) c. \(\mathbf{u}=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right] ; \mathbf{v}=\left[\begin{array}{r}-1 \\ 0 \\ 1\end{array}\right]\) d. \(\mathbf{u}=\left[\begin{array}{r}2 \\ 0 \\ -1\end{array}\right] ; \mathbf{v}=\left[\begin{array}{r}-8 \\ 0 \\ 4\end{array}\right]\)
Problem 7
Simplify \((a \mathbf{u}+b \mathbf{v}) \times(c \mathbf{u}+d \mathbf{v})\).
Problem 7
Show that the triangle with vertices \(A(4,-7,9), B(6,4,4),\) and \(C(7,10,-6)\) is not a rightangled triangle.
Problem 8
If \(A\) is \(3 \times 3\) and invertible, show that the image of the plane through the origin with normal \(\mathbf{n}\) is the plane through the origin with normal \(\mathbf{n}_{1}=B \mathbf{n}\) where \(B=\left(A^{-1}\right)^{T}\). [Hint: Use the fact that \(\mathbf{v} \cdot \mathbf{w}=\mathbf{v}^{T} \mathbf{w}\) to show that \(\mathbf{n}_{1} \cdot(A \mathbf{p})=\mathbf{n} \cdot \mathbf{p}\) for each \(\mathbf{p}\) in \(\mathbb{R}^{3}\).]
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 \(C(4,3,-3)\)
