91Ó°ÊÓ

Show that the point \(P(3,1,2)\) is equidistant from the points \(A(2,-1,3)\) and \(B(4,3,1)\)

Find an equation for the set of all points equidistant from the planes \(y=3\) and \(y=-1\).

Find an equation for the set of all points equidistant from the point (0,0,2) and the \(x y\) -plane.

Use the component form to generate an equation for the plane through \(P_{1}(4,1,5)\) normal to \(\mathbf{n}_{1}=\mathbf{i}-2 \mathbf{j}+\mathbf{k} .\) Then generate another equation for the same plane using the point \(P_{2}(3,-2,0)\) and the normal vector \(\mathbf{n}_{2}=-\sqrt{2} \mathbf{i}+2 \sqrt{2} \mathbf{j}-\sqrt{2} \mathbf{k}\)

Find the point on the sphere \(x^{2}+(y-3)^{2}+(z+5)^{2}=4\) nearest a. the \(x y\) -plane. b. the point (0,7,-5)

Find the points in which the line \(x=1+2 t, y=-1-t\) \(z=3 t\) meets the coordinate planes. Describe the reasoning behind your answer.

Find the point equidistant from the points (0,0,0),(0,4,0) \((3,0,0),\) and (2,2,-3).

Find equations for the line in the plane \(z=3\) that makes an angle of \(\bar{\pi} / 6\) rad with \(\mathbf{i}\) and an angle of \(\pi / 3\) rad with \(\mathbf{j}\). Describe the reasoning behind your answer.

Is the line \(x=1-2 t, y=2+5 t, z=-3 t\) parallel to the plane \(2 x+y-z=8 ?\) Give reasons for your answer.

How can you tell when two planes \(A_{1} x+B_{1} y+C_{1} z=D_{1}\) and \(A_{2} x+B_{2} y+C_{2} z=D_{2}\) are parallel? Perpendicular? Give reasons for your answer.