91Ó°ÊÓ

Use a computer algebra system to graph a view of the cylinder \(y^{2}+z^{2}=4\) from the points (a) \((10,0,0),\) (b) \((0,10,0),\) and (c) (10,10,10)

Find \((a) \mathbf{u} \times \mathbf{v},(b) \mathbf{v} \times \mathbf{u},\) and \((\mathbf{c}) \mathbf{v} \times \mathbf{v}\). $$ \begin{array}{l} \mathbf{u}=\langle 3,-2,-2\rangle \\ \mathbf{v}=\langle 1,5,1\rangle \end{array} $$

In Exercises \(9-14,\) find an equation in cylindrical coordinates for the equation given in rectangular coordinates. $$ z=x^{2}+y^{2}-2 $$

Find the angle \(\theta\) between the vectors. $$ \mathbf{u}=\cos \left(\frac{\pi}{6}\right) \mathbf{i}+\sin \left(\frac{\pi}{6}\right) \mathbf{j}, \quad \mathbf{v}=\cos \left(\frac{3 \pi}{4}\right) \mathbf{i}+\sin \left(\frac{3 \pi}{4}\right) \mathbf{j} $$

Find the area of the region. Interior of \(r=1-\sin \theta\) (above the polar axis)

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

Find sets of (a) parametric equations and (b) symmetric equations of the line through the two points. (For each line, write the direction numbers as integers.) $$ (2,3,0),(10,8,12) $$

Use a graphing utility to graph the polar equation and find the area of the given region. Inner loop of \(r=1+2 \cos \theta\)

Find the angle \(\theta\) between the vectors. $$ \begin{array}{l} \mathbf{u}=\langle 1,1,1\rangle \\ \mathbf{v}=\langle 2,1,-1\rangle \end{array} $$

In Exercises \(9-14,\) find an equation in cylindrical coordinates for the equation given in rectangular coordinates. $$ x^{2}+y^{2}+z^{2}=10 $$