91Ó°ÊÓ

An equation is given in spherical coordinates. Express the equation in rectangular coordinates and sketch the graph. $$ \rho=3 $$

Find an equation of the plane that satisfies the stated conditions. The plane through the point \((-1,4,2)\) that contains the line of intersection of the planes \(4 x-y+z-2=0\) and \(2 x+y-2 z-3=0\)

The given equation represents a quadric surface whose orientation is different from that in Table 11.7.1. Identify and sketch the surface. $$ x^{2}-3 y^{2}-3 z^{2}=0 $$

Describe the surface whose equation is given. $$ x^{2}+y^{2}+z^{2}-3 x+4 y-8 z+25=0 $$

Describe the surface whose equation is given. $$ x^{2}+y^{2}+z^{2}-2 x-6 y-8 z+1=0 $$

Suppose that \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=3 .\) Find $$ \begin{array}{ll}{\text { (a) } \mathbf{u} \cdot(\mathbf{w} \times \mathbf{v})} & {\text { (b) }(\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}} \\\ {\text { (c) } \mathbf{w} \cdot(\mathbf{u} \times \mathbf{v})} & {\text { (d) } \mathbf{v} \cdot(\mathbf{u} \times \mathbf{w})} \\\ {(\mathbf{e})(\mathbf{u} \times \mathbf{w}) \cdot \mathbf{v}} & {\text { (f) } \mathbf{v} \cdot(\mathbf{w} \times \mathbf{w})}\end{array} $$

Where does the line \(x=2-t, y=3 t, z=-1+2 t\) intersect the plane \(2 y+3 z=6 ?\)

True–False Determine whether the statement is true or false. Explain your answer. If \(\mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}\) and \(\mathbf{a} \neq \mathbf{0},\) then \(\mathbf{b}=\mathbf{c}\)

Find an equation of the plane that satisfies the stated conditions. The plane through \((-1,4,-3)\) that is perpendicular to the line \(x-2=t, y+3=2 t, z=-t\)

The given equation represents a quadric surface whose orientation is different from that in Table 11.7.1. Identify and sketch the surface. $$ x-y^{2}-4 z^{2}=0 $$