The points $P(a, b, c)$ and $Q(2,3,5)$ are symmetric in the sense given. Find $a, b, c$. About the plane $z=4$.

Restrict $t$ so that the equations

$$x(t)=7-5 t, \quad y(t)=-3+2 t, \quad z(t)=4-t$$

parametrize the line segment that begins at $P(12,-5,5)$ and ends at $Q(-3,1,2)$.

Determine whether the lines $l_1$ and $l_2$ are parallel, skew, or intersecting. If they intersect, find the point of intersection. $l_1: x=t, y=-t, z=-6+2 t ; \quad l_2: x=1-u$, $y=1+3 u, z=2 u$.

Find the unit vector in the direction of a.

$$a=i-2 j+2 k$$

Find a vector parametrization for the line segment that begins at $P(2,7,-1)$ and ends at $Q(4,2,3)$.

The points $P(a, b, c)$ and $Q(2,3,5)$ are symmetric in the sense given. Find $a, b, c$. About the plane $y=-1$.

Find the volume of the parallelepiped with the edges given.

$$\mathbf{i}+\mathbf{j}, \quad \mathbf{i}-\mathbf{k}, \quad 3 \mathbf{j}+\mathbf{k}$$

Find the direction cosines and direction angles of the vector.

$$\mathbf{a}=3 \mathbf{i}+12 \mathbf{j}+4 \mathbf{k}$$

Find an equation in $x, y, z$ for the plane that passes through the given points. $P_1(1,0,1), \quad P_2(2,1,0), \quad P_3(1,1,1)$.

Find the distance from the point to the plane. $P(1,3,4) ; \quad x+y-2 z=0$.

Find the direction cosines and direction angles of the vector.

$$\mathbf{a}=3 \mathbf{i}+5 \mathbf{j}-4 \mathbf{k}$$

Find a unit vector $\mathbf{N}$ which is perpendicular to the plane of $P,Q,R$, and find the area of triangle $PQR$.

$$P(2,-1,3), \quad Q(4,1,-1), \quad R(-3,0,5)$$

Find the unit vector in the direction of a.

$$a=-2 i+3 j$$

The points $P(3,2,-1), Q(7,-5,4), R(5,6,-3)$ are given. a. Find scalar parametric equations for the line that passes through $R$ and is parallel to the line determined by $P$ and Q. b. Find scalar parametric equations for the line that passes through $R$ and is perpendicular to the line determined by $P$ and $Q$. c. The lines in parts (a) and (b) determine a plane. Find an equation in rectangular coordinates for this plane.