Chapter 11

Find the acute angles between the intersecting lines. $$x=t, y=2 t, z=-t \quad \text { and } \quad x=1-t, y=5+t, z=2 t$$

Write inequalities to describe the sets. The (a) interior and (b) exterior of the sphere of radius 1 centered at the point (1,1,1)

Find a concise \(3 \times 3\) determinant formula that gives the area of a triangle in the \(x y\) -plane having vertices \(\left(a_{1}, a_{2}\right),\left(b_{1}, b_{2}\right),\) and \(\left(c_{1}, c_{2}\right)\).

Plot the surfaces in Exercises over the indicated domains. If you can, rotate the surface into different viewing positions. $$z=1-y^{2}, \quad-2 \leq x \leq 2, \quad-2 \leq y \leq 2$$

Find the acute angles between the intersecting lines. $$\begin{aligned}&x=2+t, y=4 t+2, z=1+t \quad \text { and }\\\&x=3 t-2, y=-2, z=2-2 t\end{aligned}$$

Write inequalities to describe the sets. The closed region bounded by the spheres of radius 1 and radius 2 centered at the origin. (Closed means the spheres are to be included. Had we wanted the spheres left out, we would have asked for the open region bounded by the spheres. This is analogous to the way we use closed and open to describe intervals: closed means endpoints included, open means endpoints left out. Closed sets include boundaries; open sets leave them out.)

Using the methods of Section \(6.1,\) where volume is computed by integrating cross-sectional area, it can be shown that the volume of a tetrahedron formed by three vectors is equal to \(\frac{1}{6}\) the volume of the parallelipiped formed by the three vectors. Find the volumes of the tetrahedra whose vertices are given. $$A(0,0,0), \quad B(2,0,0), \quad C(0,3,0), \quad D(0,0,4)$$

Plot the surfaces in Exercises over the indicated domains. If you can, rotate the surface into different viewing positions. $$z=x^{2}+y^{2}, \quad-3 \leq x \leq 3, \quad-3 \leq y \leq 3$$

Location \(\quad\) A bird flies from its nest \(5 \mathrm{km}\) in the direction \(60^{\circ}\) north of east, where it stops to rest on a tree. It then flies \(10 \mathrm{km}\) in the direction due southeast and lands atop a telephone pole. Place an \(x y\) -coordinate system so that the origin is the bird's nest, the \(x\) -axis points east, and the \(y\) -axis points north. a. At what point is the tree located? b. At what point is the telephone pole?

Find the center \(C\) and the radius \(a\) for the spheres. $$(x+2)^{2}+y^{2}+(z-2)^{2}=8$$

Plot the surfaces in Exercises over the indicated domains. If you can, rotate the surface into different viewing positions. \(z=x^{2}+2 y^{2} \quad\) over a. \(-3 \leq x \leq 3,-3 \leq y \leq 3\) b. \(-1 \leq x \leq 1,-2 \leq y \leq 3\) c. \(-2 \leq x \leq 2,-2 \leq y \leq 2\) d. \(-2 \leq x \leq 2, \quad-1 \leq y \leq 1\)

Use similar triangles to find the coordinates of the point \(Q\) that divides the segment from \(P_{1}\left(x_{1}, y_{1}, z_{1}\right)\) to \(P_{2}\left(x_{2}, y_{2}, z_{2}\right)\) into two lengths whose ratio is \(p / q=r\).

Find the acute angles between the lines and planes. $$x=2, y=3+2 t, z=1-2 t ; \quad x-y+z=0$$

Find the center \(C\) and the radius \(a\) for the spheres. $$(x-1)^{2}+\left(y+\frac{1}{2}\right)^{2}+(z+3)^{2}=25$$

Using the methods of Section \(6.1,\) where volume is computed by integrating cross-sectional area, it can be shown that the volume of a tetrahedron formed by three vectors is equal to \(\frac{1}{6}\) the volume of the parallelipiped formed by the three vectors. Find the volumes of the tetrahedra whose vertices are given. $$A(1,-1,0), \quad B(0,2,-2), \quad C(-3,0,3), \quad D(0,4,4)$$

Use a CAS to plot the surfaces in Exercises. Identify the type of quadric surface from your graph. $$\frac{x^{2}}{9}+\frac{y^{2}}{36}=1-\frac{z^{2}}{25}$$

Medians of a triangle \(\quad\) Suppose that \(A, B,\) and \(C\) are the comer points of the thin triangular plate of constant density shown here. a. Find the vector from \(C\) to the midpoint \(M\) of side \(A B\). b. Find the vector from \(C\) to the point that lies two-thirds of the way from \(C\) to \(M\) on the median \(C M\). c. Find the coordinates of the point in which the medians of \(\Delta A B C\) intersect. According to Exercise \(19 .\) Section \(6.6,\) this point is the plate's center of mass. (See the figure.) Figure cannot copy

Use a calculator to find the acute angles between the planes to the nearest hundredth of a radian. $$2 x+2 y+2 z=3, \quad 2 x-2 y-z=5$$

Find the center \(C\) and the radius \(a\) for the spheres. $$(x-\sqrt{2})^{2}+(y-\sqrt{2})^{2}+(z+\sqrt{2})^{2}=2$$

Using the methods of Section \(6.1,\) where volume is computed by integrating cross-sectional area, it can be shown that the volume of a tetrahedron formed by three vectors is equal to \(\frac{1}{6}\) the volume of the parallelipiped formed by the three vectors. Find the volumes of the tetrahedra whose vertices are given. $$A(-1,2,3), \quad B(2,0,1), \quad C(1,-3,2), \quad D(-2,1,-1)$$

Use a CAS to plot the surfaces in Exercises. Identify the type of quadric surface from your graph. $$\frac{x^{2}}{9}-\frac{z^{2}}{9}=1-\frac{y^{2}}{16}$$

Find the vector from the origin to the point of intersection of the medians of the triangle whose vertices are $$A(1,-1,2), \quad B(2,1,3), \quad \text { and } \quad C(-1,2,-1).$$

Use a calculator to find the acute angles between the planes to the nearest hundredth of a radian. $$x+y+z=1, \quad z=0 \quad \text { (the } x y \text { -plane) }$$

Find the center \(C\) and the radius \(a\) for the spheres. $$x^{2}+\left(y+\frac{1}{3}\right)^{2}+\left(z-\frac{1}{3}\right)^{2}=\frac{16}{9}$$

Determine whether the given points are coplanar. $$A(1,1,1), \quad B(-1,0,4), \quad C(0,2,1), \quad D(2,-2,3)$$

Let \(A B C D\) be a general, not necessarily planar, quadrilateral in space. Show that the two segments joining the midpoints of opposite sides of \(A B C D\) bisect each other. (Hint: Show that the segments have the same midpoint.)

Use a calculator to find the acute angles between the planes to the nearest hundredth of a radian. $$2 x+2 y-z=3, \quad x+2 y+z=2$$

Find the center \(C\) and the radius \(a\) for the spheres. $$x^{2}+y^{2}+z^{2}+4 x-4 z=0$$

Determine whether the given points are coplanar. $$A(0,0,4), \quad B(6,2,0), \quad C(2,-1,1), \quad D(-3,-4,3)$$

Use a CAS to plot the surfaces in Exercises. Identify the type of quadric surface from your graph. $$\frac{y^{2}}{16}=1-\frac{x^{2}}{9}+z$$

Vectors are drawn from the center of a regular \(n\) -sided polygon in the plane to the vertices of the polygon. Show that the sum of the vectors is zero. (Hint: What happens to the sum if you rotate the polygon about its center?)

Use a calculator to find the acute angles between the planes to the nearest hundredth of a radian. $$4 y+3 z=-12, \quad 3 x+2 y+6 z=6$$

Find the center \(C\) and the radius \(a\) for the spheres. $$x^{2}+y^{2}+z^{2}-6 y+8 z=0$$

Determine whether the given points are coplanar. $$A(0,1,2), \quad B(-1,1,0), \quad C(2,0,-1), \quad D(1,-1,1)$$

Use a CAS to plot the surfaces in Exercises. Identify the type of quadric surface from your graph. $$\frac{x^{2}}{9}-1=\frac{y^{2}}{16}+\frac{z^{2}}{2}$$

Suppose that \(A, B,\) and \(C\) are vertices of a triangle and that \(a, b\) and \(c\) are, respectively, the midpoints of the opposite sides. Show that \(\overrightarrow{A a}+\overrightarrow{B b}+\overrightarrow{C c}=0\).

find the point in which the line meets the plane. $$x=1-t, \quad y=3 t, \quad z=1+t ; \quad 2 x-y+3 z=6$$

Find the center \(C\) and the radius \(a\) for the spheres. $$2 x^{2}+2 y^{2}+2 z^{2}+x+y+z=9$$

Use a CAS to plot the surfaces in Exercises. Identify the type of quadric surface from your graph. $$y-\sqrt{4-z^{2}}=0$$

Unit vectors in the plane \(\quad\) Show that a unit vector in the plane can be expressed as \(\mathbf{u}=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j},\) obtained by rotating \(i\) through an angle \(\theta\) in the counterclockwise direction. Explain why this form gives every unit vector in the plane.

find the point in which the line meets the plane. $$x=2, \quad y=3+2 t, \quad z=-2-2 t ; \quad 6 x+3 y-4 z=-12$$

Find the center \(C\) and the radius \(a\) for the spheres. $$3 x^{2}+3 y^{2}+3 z^{2}+2 y-2 z=9$$

Consider a triangle whose vertices are \(A(2,-3,4), B(1,0,-1)\) and \(C(3,1,2)\). a. Find \(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}\). b. Find \(\overrightarrow{B A}+\overrightarrow{A C}+\overrightarrow{C B}\).

find the point in which the line meets the plane. $$x=1+2 t, \quad y=1+5 t, \quad z=3 t ; \quad x+y+z=2$$

Find the center \(C\) and the radius \(a\) for the spheres. $$x^{2}+y^{2}+z^{2}-4 x+6 y-10 z=11$$

find the point in which the line meets the plane. $$x=-1+3 t, \quad y=-2, \quad z=5 t ; \quad 2 x-3 z=7$$

Find the center \(C\) and the radius \(a\) for the spheres. $$(x-1)^{2}+(y-2)^{2}+(z+1)^{2}=103+2 x+4 y-2 z$$

Find equations for the spheres whose centers and radii are given in Exercises \(61-64\). $$ \begin{array}{lc} \text { Center } & \text { Radius } \\ \hline(1,2,3) & \sqrt{14} \end{array} $$

Find parametrizations for the lines in which the planes. $$3 x-6 y-2 z=3, \quad 2 x+y-2 z=2$$

Find equations for the spheres whose centers and radii are given. $$ (0,-1,5) \quad 2 $$