91Ó°ÊÓ

Sketch the graph of the quadric surface. Hyperboloids of two sheets (a) \(x^{2}-\frac{y^{2}}{4}-z^{2}=1\) (b) \(\frac{z^{2}}{4}-y^{2}-x^{2}=1\)

Find an equation of the plane that satisfies the stated conditions. Through \(P(-4,1,6)\) and having the same trace in the \(x z\) -plane as the plane \(x+4 y-5 z=8\)

Prove that \((\mathrm{a} \times \mathbf{b}) \cdot \mathbf{b}=0\) for all vectors \(\mathbf{a}\) and \(\mathbf{b}\).

Sketch the graph of the quadric surface. Hyperboloids of two sheets (a) \(z^{2}-\frac{x^{2}}{4}-\frac{y^{2}}{4}=1\) (b) \(\frac{y^{2}}{4}-x^{2}-\frac{z^{2}}{9}=1\)

If the vector a represents a constant force, find the work done when its point of application moves along the line segment from \(P\) to \(Q\). \(\mathbf{a}=\langle 8,0,-4\rangle ; \quad P(-1,2,5), \quad Q(4,1,0)\)

Find a symmetric form for the line \(x=3+4 t, y=2-t\). \(z=2 t\)

Find an equation of the sphere that has endpoints of a diameter at \(A\) and \(B\). $$ A(4,-3,4), \quad B(0,3,-10) $$

Find an equation of the plane that satisfies the stated conditions. $$ \begin{aligned} &\text { Through the origin and the points } P(0,2,5) \text { and }\\\ &Q(1,4,0) \end{aligned} $$

If \(\mathbf{a} \times \mathbf{b}=\mathbf{a} \times \mathbf{c}\) and \(\mathbf{a} \neq \mathbf{0},\) does it follow that \(\mathbf{b}=\mathbf{c}\) ? Explain.

Find an equation of the plane through \(P\), \(Q,\) and \(R\) $$ P(1,1,3), \quad Q(-1,3,2), \quad R(1,-1,2) $$