Problem 26
Show that every line parallel to the \(z\) axis has parametric equations \(x=x_{0}, y=y_{0}, z=t\) for some fixed numbers \(x_{0}\) and \(y_{0}\).
Problem 27
Let \(\mathbf{d}=\left[\begin{array}{l}a \\ b \\ c\end{array}\right]\) be a vector where \(a\) \(b,\) and \(c\) are all nonzero. Show that the equations of the line through \(P_{0}\left(x_{0}, y_{0}, z_{0}\right)\) with direction vector \(\mathbf{d}\) can be written in the form $$\frac{x-x_{0}}{a}=\frac{y-y_{0}}{b}=\frac{z-z_{0}}{c}$$ This is called the symmetric form of the equations.
Problem 27
Given a rectangular solid with sides of lengths \(1,1,\) and \(\sqrt{2}\), find the angle between a diagonal and one of the longest sides.
Problem 28
Consider a rectangular solid with sides of lengths \(a, b,\) and \(c .\) Show that it has two orthogonal diagonals if and only if the sum of two of \(a^{2}, b^{2},\) and \(c^{2}\) equals the third.
Problem 29
Find all points \(C\) on the line through \(A(1,-1,2)\) and \(B=(2,0,1)\) such that \(\|\overrightarrow{A C}\|=2\|\overrightarrow{B C}\|\).
Problem 30
If the diagonals of a parallelogram have equal length, show that the parallelogram is a rectangle.
Problem 30
Let \(A, B, C, D, E,\) and \(F\) be the vertices of a regular hexagon, taken in order. Show that \(\overrightarrow{A B}+\overrightarrow{A C}+\overrightarrow{A D}+\overrightarrow{A E}+\overrightarrow{A F}=3 \overrightarrow{A D}\).
Problem 31
Given \(\mathbf{v}=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]\) in component form, show that the projections of \(\mathbf{v}\) on \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\) are \(x \mathbf{i}, y \mathbf{j},\) and zk, respectively.
Problem 31
a. Let \(P_{1}, P_{2}, P_{3}, P_{4}, P_{5},\) and \(P_{6}\) be six points equally spaced on a circle with centre \(C\). Show that $$ \overrightarrow{C P}_{1}+\overrightarrow{C P}_{2}+\overrightarrow{C P}_{3}+\overrightarrow{C P}_{4}+\overrightarrow{C P}_{5}+\overrightarrow{C P}_{6}=\mathbf{0} $$ b. Show that the conclusion in part (a) holds for any even set of points evenly spaced on the circle. c. Show that the conclusion in part (a) holds for three points. d. Do you think it works for any finite set of points evenly spaced around the circle?
Problem 32
Consider a quadrilateral with vertices \(A, B, C,\) and \(D\) in order (as shown in the diagram). If the diagonals \(A C\) and \(B D\) bisect each other, show that the quadrilateral is a parallelogram. (This is the converse of Example \(4.1 .2 .)\) [Hint: Let \(E\) be the intersection of the diagonals. Show that \(\overrightarrow{A B}=\overrightarrow{D C}\) by writing \(\overrightarrow{A B}=\overrightarrow{A E}+\overrightarrow{E B} \cdot]\)
