91Ó°ÊÓ

Find a symmetric form for the line througl \(P_{\text {and }} P_{2}\) $$ P_{1}(4,2,-3), \quad P_{2}(-3,2,5) $$

Exer. \(39-50:\) Sketch the graph of the equation. $$ z^{2}-4 x^{2}=9-4 y^{2} $$

Sketch the graph of the equation in an \(x y z-\) coordinate system, and identify the surface. $$y^{2}-9 x^{2}-z^{2}-9=0$$

Solve without using components for the vectors. Prove that \(\|\mathrm{a}+\mathrm{b}\|^{2}+\|\mathrm{a}-\mathrm{b}\|^{2}=2\left(\|\mathrm{a}\|^{2}+\|\mathrm{b}\|^{2}\right) .\)

Exer. \(39-50:\) Sketch the graph of the equation. $$ 2 x^{2}+4 z^{2}-y^{2}=0 $$

Find a symmetric form for the line througl \(P_{\text {and }} P_{2}\) $$ P_{1}(5,-7,4), \quad P_{2}(-2,-1,4) $$

(a) Let a \(=\left\langle a_{1}, a_{2}\right\rangle\) be any nonzero vector, and let \(\overrightarrow{O A}\) be the position vector for a. If \(\theta\) is the smallest nonnegative angle from the positive \(x\) -axis to \(\overrightarrow{O A}\), show that \(\mathbf{a}=\|\mathbf{a}\|(\cos \theta \mathbf{i}+\sin \theta \mathbf{j})\). (b) Show that every unit vector in \(V_{2}\) can be expressed in the form \(\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) for some \(\theta\)

Sketch the graph of the equation in an \(x y z-\) coordinate system, and identify the surface. $$36 x^{2}-16 y^{2}+9 z^{2}=0$$

Let \(\mathbf{r}_{0}=\left\langle x_{0}, y_{0}\right\rangle, \mathbf{r}=\langle x, y\rangle,\) and \(c>0 .\) Describe the set of all points \(P(x, y)\) such that \(\left\|\mathbf{r}-\mathbf{r}_{0}\right\|=c\).

Find parametric equations for the line of intersection of the two planes. $$ x+2 y-9 z=7, \quad 2 x-3 y+17 z=0 $$