91Ó°ÊÓ

Find an equation in cylindrical coordinates for the equation given in rectangular coordinates. \(x^{2}+y^{2}+z^{2}-3 z=0\)

The line passes through the point \((-6,0,8)\) and is parallel to the line \(x=5-2 t, y=-4+2 t, z=0\).

Determine the location of a point \((x, y, z)\) that satisfies the condition(s). \(x y>0, \quad z=-3\)

In Exercises \(21-24\), find the coordinates of a point \(P\) on the line and a vector \(\mathbf{v}\) parallel to the line. \(x=3-t, \quad y=-1+2 t, \quad z=-2\)

Use a computer algebra system to find \(\mathbf{u} \times \mathbf{v}\) and a unit vector orthogonal to \(\mathbf{u}\) and \(\mathbf{v}\). $$ \begin{aligned} &\mathbf{u}=\langle 4,-3.5,7\rangle \\ &\mathbf{v}=\langle-1,8,4\rangle \end{aligned} $$

Determine whether \(\mathrm{u}\) and \(\mathrm{v}\) are orthogonal, parallel, or neither. $$ \begin{aligned} &\mathbf{u}=\langle 4,3\rangle \\ &\mathbf{v}=\left\langle\frac{1}{2},-\frac{2}{3}\right\rangle \end{aligned} $$

Identify and sketch the quadric surface. Use a computer algebra system to confirm your sketch. $$ 16 x^{2}-y^{2}+16 z^{2}=4 $$

Find the coordinates of a point \(P\) on the line and a vector \(\mathbf{v}\) parallel to the line.\(x=4 t, \quad y=5-t, \quad z=4+3 t\)

Identify and sketch the quadric surface. Use a computer algebra system to confirm your sketch. $$ z^{2}-x^{2}-\frac{y^{2}}{4}=1 $$

Determine the location of a point \((x, y, z)\) that satisfies the condition(s). \(x y<0, \quad z=4\)