91Ó°ÊÓ

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

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

Identify and sketch the quadric surface. Use a computer algebra system to confirm your sketch. $$ x^{2}-y+z^{2}=0 $$

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

Find (a) \(\frac{2}{3} \mathrm{u}\), (b) \(\mathrm{v}-\mathrm{u}\), and \((\mathrm{c}) 2 \mathrm{u}+5 \mathrm{v}\) $$ \begin{aligned} &\mathbf{u}=\langle 4,9\rangle \\ &\mathbf{v}=\langle 2,-5\rangle \end{aligned} $$

Determine whether \(\mathrm{u}\) and \(\mathrm{v}\) are orthogonal, parallel, or neither. $$ \begin{aligned} &\mathbf{u}=\mathbf{j}+6 \mathbf{k} \\ &\mathbf{v}=\mathbf{i}-2 \mathbf{j}-\mathbf{k} \end{aligned} $$

Find an equation in rectangular coordinates for the equation given in cylindrical coordinates, and sketch its graph. \(\theta=\pi / 6\)

Use a computer algebra system to find \(\mathbf{u} \times \mathbf{v}\) and a unit vector orthogonal to \(\mathbf{u}\) and \(\mathbf{v}\). $$ \begin{aligned} &\mathbf{u}=\frac{2}{3} \mathbf{k} \\ &\mathbf{v}=\frac{1}{2} \mathbf{i}+\mathbf{6 k} \end{aligned} $$

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