Chapter 11

Find parametrizations for the lines in which the planes. $$x-2 y+4 z=2, \quad x+y-2 z=5$$

Find equations for the spheres whose centers and radii are given. $$\begin{array}{l}\text { (Radius) } \\\\\sqrt{14} \\\2 \\\\\frac{4}{9} \\\7\end{array}$$ (Center) $$\left(-1, \frac{1}{2},-\frac{2}{3}\right)$$

Find parametrizations for the lines in which the planes. $$5 x-2 y=11, \quad 4 y-5 z=-17$$

Find equations for the spheres whose centers and radii are given $$ \begin{array}{ll} \text { Center } & \text { Radius } \\ (0,-7,0) & 7 \end{array} $$

Given two lines in space, either they are parallel, they intersect, or they are skew (lie in parallel planes). Determine whether the lines, taken two at a time, are parallel, intersect, or are skew. If they intersect, find the point of intersection. Otherwise, find the distance between the two lines. $$\begin{array}{ll}L 1: x=3+2 t, y=-1+4 t, z=2-t ; & -\infty

Find a formula for the distance from the point \(P(x, y, z)\) to the a. \(x\) -axis. b. \(y\) -axis. c. z-axis.

Given two lines in space, either they are parallel, they intersect, or they are skew (lie in parallel planes). Determine whether the lines, taken two at a time, are parallel, intersect, or are skew. If they intersect, find the point of intersection. Otherwise, find the distance between the two lines. $$\begin{aligned}&L 1: x=1+2 t, \quad y=-1-t, \quad z=3 t ; \quad-\infty

Find a formula for the distance from the point \(P(x, y, z)\) to the a. \(x y\) -plane. b. \(y z\) -plane. c. \(x z\) -plane.

Use Equations ( 3 ) to generate a parametrization of the line through \(P(2,-4,7)\) parallel to \(\mathbf{v}_{1}=2 \mathbf{i}-\mathbf{j}+3 \mathbf{k} .\) Then generate another parametrization of the line using the point \(P_{2}(-2,-2,1)\) and the vector \(\mathbf{v}_{2}=-\mathbf{i}+(1 / 2) \mathbf{j}-(3 / 2) \mathbf{k}\)

Find the perimeter of the triangle with vertices \(A(-1,2,1)\) \(B(1,-1,3),\) and \(C(3,4,5)\)

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 \(\mathbf{n}_{2}=-\sqrt{2} \mathbf{i}+2 \sqrt{2} \mathbf{j}-\sqrt{2} \mathbf{k}\)

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

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

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

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

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.

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)

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.

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

Find two different planes whose intersection is the line \(x=1+t, y=2-t, z=3+2 t .\) Write equations for each plane in the form \(A x+B y+C z=D\)

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

Find a plane through the origin that is perpendicular to the plane \(M: 2 x+3 y+z=12\) in a right angle. How do you know that your plane is perpendicular to \(M ?\)

Find an equation for the set of points equidistant from the \(y\) -axis and the plane \(z=6\)

The graph of \((x / a)+(y / b)+(z / c)=1\) is a plane for any nonzero numbers \(a, b,\) and \(c .\) Which planes have an equation of this form?

Find an equation for the set of points equidistant from the a. \(x y\) -plane and the \(y z\) -plane. b. \(x\) -axis and the \(y\) -axis.

Suppose \(L_{1}\) and \(L_{2}\) are disjoint (nonintersecting) nonparallel lines. Is it possible for a nonzero vector to be perpendicular to both \(L_{1}\) and \(L_{2} ?\) Give reasons for your answer.

\(.\) Find all points that simultaneously lie 3 units from each of the points \((2,0,0),(0,2,0),\) and (0,0,2)