91Ó°ÊÓ

Find parametric equations and symmetric equations for the line. The line through \((2,1,0)\) and perpendicular to both \(\mathbf{i}+\mathbf{j}\) and \(\mathbf{j}+\mathbf{k}\)

Find the distance from \((3,7,-5)\) to each of the following. \(\begin{array}{ll}{\text { (a) The } x y \text { -plane }} & {\text { (b) The } y z \text { -plane }} \\ {\text { (c) The } x z \text { -plane }} & {\text { (d) The } x \text { -axis }} \\ {\text { (e) The } y \text { -axis }} & {\text { (f) The } z \text { -axis }}\end{array}\)

Find a vector a with representation given by the directed line segment \(\overline{A B}\) . Draw \(\overline{A B}\) and the equivalent representation starting at the origin. $$A(2,1), \quad B(0,6)$$

(a) Find and identify the traces of the quadric surface \(-x^{2}-y^{2}+z^{2}=1\) and explain why the graph looks like the graph of the hyperboloid of two sheets in Table \(1 .\) (b) If the equation in part (a) is changed to \(x^{2}-y^{2}-z^{2}=1\) \(\quad\) what happens to the graph? Sketch the new graph.

Find the vector, not with determinants, but by using properties of cross products. $$\mathbf{k} \times(\mathbf{i}-2 \mathbf{j})$$

\(3-10\) Find \(\mathbf{a} \cdot \mathbf{b}\) $$|\mathbf{a}|=3, \quad|\mathbf{b}|=\sqrt{6}, \quad$$ the angle between a and \(\mathbf{b}\) is \(45^{\circ}\)

Find the vector, not with determinants, but by using properties of cross products. $$(\mathbf{j}-\mathbf{k}) \times(\mathbf{k}-\mathbf{i})$$

Find a vector a with representation given by the directed line segment \(\overline{A B}\) . Draw \(\overline{A B}\) and the equivalent representation starting at the origin. $$A(0,3,1), \quad B(2,3,-1)$$

Find parametric equations and symmetric equations for the line. The line through \((1,-1,1)\) and parallel to the line \(x+2=\frac{1}{2} y=z-3\)

\(11-20\) Use traces to sketch and identify the surface. $$x=y^{2}+4 z^{2}$$