91Ó°ÊÓ

If \(\mathbf{r}=\langle x, y, z\rangle, \mathbf{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle,\) and \(\mathbf{b}=\left\langle b_{1}, b_{2}, b_{3}\right\rangle,\) show that the vector equation \((\mathbf{r}-\mathbf{a}) \cdot(\mathbf{r}-\mathbf{b})=0\) represents a sphere, and find its center and radius.

Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place). $$ x-y+3 z=1, \quad 3 x+y-z=2 $$

Find the angle between a diagonal of a cube and one of its edges.

Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place). $$ 2 x-3 y=z, \quad 4 x=3+6 y+2 z $$

Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place). $$ 5 x+2 y+3 z=2, \quad y=4 x-6 z $$

Find the angle between a diagonal of a cube and a diagonal of one of its faces.

\(57-58\) (a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes. $$ x+y+z=1, \quad x+2 y+2 z=1 $$

(a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes. $$ 3 x-2 y+z=1, \quad 2 x+y-3 z=3 $$

If \(\mathbf{c}=|\mathbf{a}| \mathbf{b}+|\mathbf{b}| \mathbf{a},\) where \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are all nonzero vectors, show that \(\mathbf{c}\) bisects the angle between a and \(\mathbf{b} .\)

\(59-60\) Find symmetric equations for the line of intersection of the planes. $$ 5 x-2 y-2 z=1, \quad 4 x+y+z=6 $$