91Ó°ÊÓ

$$\begin{aligned}&\text { Find the angle } A B C \text { if the points are } A(1,2,3) \text { , }\\\&B(-4,5,6), \text { and } C(1,0,1)\end{aligned}$$

Show that the triangle \(A B C\) is a right triangle if the ver tices are \(A(6,3,3), B(3,1,-1),\) and \(C(-1,10,-2.5) .\)

In Problems \(17-24,\) sketch the graphs of the given equations. Begin by sketching the traces in the coordinate planes (see Examples 4 and 5 ) $$ 3 x-4 y+2 z=24 $$

Mark pushes on a post in the direction \(\mathrm{S} 30^{\circ} \mathrm{E}\left(30^{\circ}\right.\) east of south) with a force of 60 pounds. Dan pushes on the same post in the direction \(\mathrm{S} 60^{\circ} \mathrm{W}\) with a force of 80 pounds. What are the magnitude and direction of the resultant force?

If \(\mathbf{r}(t)=\sin 2 t \mathbf{i}+\cosh t \mathbf{j}\) and \(h(t)=\ln (3 t-2),\) find \(D_{f}[h(t) \mathbf{r}(t)]\)

Show that the lines $$ \frac{x-1}{-4}=\frac{y-2}{3}=\frac{z-4}{-2} $$ and $$ \frac{x-2}{-1}=\frac{y-1}{1}=\frac{z+2}{6} $$ intersect, and find the equation of the plane that they determine.

Find symmetric equations of the line through (4,5,8) and perpendicular to the plane \(3 x+5 y+2 z=30 .\) Sketch the plane and the line.

Find the equation of the plane through (2,-1,4) that is perpendicular to both the planes \(x+y+z=2\) and \(x-y-z=4\)

Make the required change in the given equation. \(x^{2}+y^{2}+4 z^{2}=10\) to cylindrical coordinates

Find the equation of the plane through (2,-3,2) and parallel to the plane of the vectors \(4 \mathbf{i}+3 \mathbf{j}-\mathbf{k}\) and \(2 \mathbf{i}-5 \mathbf{j}+6 \mathbf{k}\).