91Ó°ÊÓ

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

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

Use a calculator to find the acute angles between the planes to the nearest hundredth of a radian. $$x+y+z=1, \quad z=0 \quad \text { (the } x y \text { -plane) }$$

Find the center \(C\) and the radius \(a\) for the spheres. $$x^{2}+\left(y+\frac{1}{3}\right)^{2}+\left(z-\frac{1}{3}\right)^{2}=\frac{16}{9}$$

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.)

Determine whether the given points are coplanar. $$A(1,1,1), \quad B(-1,0,4), \quad C(0,2,1), \quad D(2,-2,3)$$

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

Find the center \(C\) and the radius \(a\) for the spheres. $$x^{2}+y^{2}+z^{2}+4 x-4 z=0$$

Find the center \(C\) and the radius \(a\) for the spheres. $$x^{2}+y^{2}+z^{2}-6 y+8 z=0$$

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