91Ó°ÊÓ

Use a calculator to find the acute angles between the planes in Exercises \(49-52\) to the nearest hundredth of a radian. $$ 2 x+2 y-z=3, \quad x+2 y+z=2 $$

Use a calculator to find the acute angles between the planes in Exercises \(49-52\) to the nearest hundredth of a radian. $$ 4 y+3 z=-12, \quad 3 x+2 y+6 z=6 $$

Plot the surfaces in Exercises \(49-52\) over the indicated domains. If you can, rotate the surface into different viewing positions. $$z=x^{2}+2 y^{2} \quad\text{ over }$$ \begin{equation}\begin{array}{l}{\text { a. }-3 \leq x \leq 3,-3 \leq y \leq 3} \\ {\text { b. }-1 \leq x \leq 1, \quad-2 \leq y \leq 3} \\ {\text { c. }-2 \leq x \leq 2, \quad-2 \leq y \leq 2} \\ {\text { d. }-2 \leq x \leq 2, \quad-1 \leq y \leq 1}\end{array}\end{equation}

Find equations for the spheres whose centers and radii are given in Exercises \(51-54 .\) $$\frac{\text { Center }}{(0,-1,5)} \frac{\text { Radius }}{2}$$

Find the vector from the origin to the point of intersection of the medians of the triangle whose vertices are $$A(1,-1,2), \quad B(2,1,3), \quad\( and \)\quad C(-1,2,-1).$$

In Exercises \(53-56,\) find the point in which the line meets the plane. $$ x=1-t, \quad y=3 t, \quad z=1+t ; \quad 2 x-y+3 z=6 $$

Find equations for the spheres whose centers and radii are given in Exercises \(51-54 .\) $$\frac{\text { Center }}{\left(-1, \frac{1}{2},-\frac{2}{3}\right)} \frac{\text { Radius }}{\frac{4}{9}}$$

Let \(A B C D\) be a general, not necessarily planar, quadrilateral in space. Show that the two segments joining the midpoints of opposite sides of \(A B C D\) bisect each other. (Hint: Show that the segments have the same midpoint.)

In Exercises \(53-56,\) find the point in which the line meets the plane. $$ x=2, \quad y=3+2 t, \quad z=-2-2 t ; \quad 6 x+3 y-4 z=-12 $$

Use a CAS to plot the surfaces in Exercises \(53-58 .\) Identify the type of quadric surface from your graph. $$\frac{x^{2}}{9}-\frac{z^{2}}{9}=1-\frac{y^{2}}{16}$$