Chapter 12

In Exercises \(19-22,\) verify that \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=(\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}=\) \((\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v}\) and find the volume of the parallelepiped (box) determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w} .\) $$ \mathbf{i}-\mathbf{j}+\mathbf{k} \quad 2 \mathbf{i}+\mathbf{j}-2 \mathbf{k} \quad-\mathbf{i}+2 \mathbf{j}-\mathbf{k} $$

Sum of vectors \(\mathbf{u}=\mathbf{i}+(\mathbf{j}+\mathbf{k})\) is already the sum of a vector parallel to \(\mathbf{i}\) and a vector orthogonal to \(\mathbf{i}\) . If you use \(\mathbf{v}=\mathbf{i},\) in the decomposition \(\mathbf{u}=\operatorname{proj}_{v} \mathbf{u}+\left(\mathbf{u}-\operatorname{proj}_{v} \mathbf{u}\right),\) do you get projy \(\mathbf{u}=\mathbf{i}\) and \(\left(\mathbf{u}-\operatorname{proj}_{\mathbf{v}} \mathbf{u}\right)=\mathbf{j}+\mathbf{k} ?\) Try it and find out.

In Exercises \(19-28\) , describe the given set with a single equation or with a pair of equations. The plane through the point \((3,-1,2)\) perpendicular to the a. \(x\) -axis \(\quad\) b. \(y\) -axis \(\quad\) c. \(z\) -axis

In Exercises \(17-22,\) express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k} .\) \(\overrightarrow{A B}\) if \(A\) is the point \((1,0,3)\) and \(B\) is the point \((-1,4,5)\)

In Exercises \(19-22,\) verify that \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=(\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}=\) \((\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v}\) and find the volume of the parallelepiped (box) determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w} .\) $$ 2 \mathbf{i}+\mathbf{j} \quad 2 \mathbf{i}-\mathbf{j}+\mathbf{k} \quad \mathbf{i}+2 \mathbf{k} $$

Find equations for the planes in Exercises 21–26. The plane through \(P_{0}(0,2,-1)\) normal to \(\mathbf{n}=3 \mathbf{i}-2 \mathbf{j}-\mathbf{k}\)

In Exercises \(19-28\) , describe the given set with a single equation or with a pair of equations. The plane through the point \((3,-1,1)\) parallel to the a. \(x\) y-plane \(\quad\) b. \(y z\) -plane \(\quad\) c. \(x z\) -plane

In Exercises \(17-22,\) express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k} .\) $$ 5 \mathbf{u}-\mathbf{v} \text { if } \mathbf{u}=\langle 1,1,-1\rangle \text { and } \mathbf{v}=\langle 2,0,3\rangle $$

In Exercises \(19-22,\) verify that \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=(\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}=\) \((\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v}\) and find the volume of the parallelepiped (box) determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w} .\) $$ \mathbf{i}+\mathbf{j}-2 \mathbf{k} \quad-\mathbf{i}-\mathbf{k} \quad 2 \mathbf{i}+4 \mathbf{j}-2 \mathbf{k} $$

Find equations for the planes in Exercises 21–26. The plane through \((1,-1,3)\) parallel to the plane $$3 x+y+z=7$$

Orthogonality on a circle Suppose that \(A B\) is the diameter of a circle with center \(O\) and that \(C\) is a point on one of the two arcs joining \(A\) and \(B\) . Show that \(\overrightarrow{C A}\) and \(\overline{C B}\) are orthogonal.

In Exercises \(19-28\) , describe the given set with a single equation or with a pair of equations. The circle of radius 2 centered at \((0,0,0)\) and lying in the a. \(x y\) -plane \(\quad\) b. \(y z\) -plane \(\quad\) c. \(x\) -plane

In Exercises \(17-22,\) express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k} .\) $$ -2 \mathbf{u}+3 \mathbf{v} \text { if } \mathbf{u}=\langle- 1,0,2\rangle \text { and } \mathbf{v}=\langle 1,1,1\rangle $$

Sketch the surfaces in Exercises \(13-76\) $$ 4 x^{2}+9 y^{2}+4 z^{2}=36 $$

Parallel and perpendicular vectors Let \(\mathbf{u}=5 \mathbf{i}-\mathbf{j}+\mathbf{k}, \mathbf{v}=\) \(\mathbf{j}-5 \mathbf{k}, \mathbf{w}=-15 \mathbf{i}+3 \mathbf{j}-3 \mathbf{k} .\) Which vectors, if any, are \((\mathbf{a})\) perpendicular? (b) Parallel? Give reasons for your answers.

Find equations for the planes in Exercises 21–26. The plane through \((1,1,-1),(2,0,2),\) and \((0,-2,1)\)

Diagonals of a rhombus Show that the diagonals of a rhombus (parallelogram with sides of equal length) are perpendicular.

In Exercises \(19-28\) , describe the given set with a single equation or with a pair of equations. The circle of radius 2 centered at \((0,2,0)\) and lying in the a. \(x y\) -plane \(\quad\) b. \(y z\) -plane \(\quad\) c. plane \(y=2\)

Parallel and perpendicular vectors Let \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}-\mathbf{k},\) \(\mathbf{v}=-\mathbf{i}+\mathbf{j}+\mathbf{k}, \mathbf{w}=\mathbf{i}+\mathbf{k}, \mathbf{r}=-(\pi / 2) \mathbf{i}-\pi \mathbf{j}+(\pi / 2) \mathbf{k}\) . Which vectors, if any, are (a) perpendicular? (b) Parallel? Give reasons for your answers.

Find equations for the planes in Exercises 21–26. The plane through \((2,4,5),(1,5,7),\) and \((-1,6,8)\)

Perpendicular diagonals Show that squares are the only rectangles with perpendicular diagonals.

In Exercises \(19-28\) , describe the given set with a single equation or with a pair of equations. The circle of radius 1 centered at \((-3,4,1)\) and lying in a plane parallel to the a. \(x y\) -plane \(\quad\) b. \(y z\) -plane \(\quad\) c. \(x z\) -plane

Find equations for the planes in Exercises 21–26. The plane through \(P_{0}(2,4,5)\) perpendicular to the line $$ x=5+t, \quad y=1+3 t, \quad z=4 t $$

When parallelograms are rectangles Prove that a parallelogram is a rectangle if and only if its diagonals are equal in length. (This fact is often exploited by carpenters.)

In Exercises \(19-28\) , describe the given set with a single equation or with a pair of equations. The line through the point \((1,3,-1)\) parallel to the a. \(x\) -axis \(\quad\) b. y-axis \(\quad\) c. \(z\) -axis

In Exercises 25–30, express each vector as a product of its length and direction. $$ 2 \mathbf{i}+\mathbf{j}-2 \mathbf{k} $$

Sketch the surfaces in Exercises \(13-76\) $$ z=x^{2}+4 y^{2} $$

Find equations for the planes in Exercises 21–26. The plane through \(A(1,-2,1)\) perpendicular to the vector from the origin to \(A\)

Diagonal of parallelogram Show that the indicated diagonal of the parallelogram determined by vectors \(\mathbf{u}\) and \(\mathbf{v}\) bisects the angle between \(\mathbf{u}\) and \(\mathbf{v}\) if \(|\mathbf{u}|=|\mathbf{v}|\)

In Exercises \(19-28\) , describe the given set with a single equation or with a pair of equations. The set of points in space equidistant from the origin and the point \((0,2,0)\)

In Exercises 25–30, express each vector as a product of its length and direction. $$ 9 \mathbf{i}-2 \mathbf{j}+6 \mathbf{k} $$

Sketch the surfaces in Exercises \(13-76\) $$ z=x^{2}+9 y^{2} $$

Which of the following are always true, and which are not always true? Give reasons for your answers. a. \(|\mathbf{u}|=\sqrt{\mathbf{u} \cdot \mathbf{u}} \quad\) b. \(\mathbf{u} \cdot \mathbf{u}=|\mathbf{u}|\) c. \(\mathbf{u} \times \mathbf{0}=\mathbf{0} \times \mathbf{u}=\mathbf{0} \quad\) d. \(\mathbf{u} \times(-\mathbf{u})=\mathbf{0}\) e. \(\mathbf{u} \times \mathbf{v}=\mathbf{v} \times \mathbf{u}\) f. \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})=\mathbf{u} \times \mathbf{v}+\mathbf{u} \times \mathbf{w}\) g. \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{v}=0\) h. \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\)

Find the point of intersection of the lines \(x=2 t+1\) \(y=3 t+2, z=4 t+3,\) and \(x=s+2, y=2 s+4, z=\) \(-4 s-1,\) and then find the plane determined by these lines.

Projectile motion A gun with muzzle velocity of 1200 \(\mathrm{ft} / \mathrm{sec}\) is fired at an angle of \(8^{\circ}\) above the horizontal. Find the horizontal and vertical components of the velocity.

In Exercises \(19-28\) , describe the given set with a single equation or with a pair of equations. The circle in which the plane through the point \((1,1,3)\) perpendicular to the z-axis meets the sphere of radius 5 centered at the origin

In Exercises 25–30, express each vector as a product of its length and direction. $$ 5 \mathbf{k} $$

Sketch the surfaces in Exercises \(13-76\) $$ z=8-x^{2}-y^{2} $$

Which of the following are always true, and which are not always true? Give reasons for your answers. \(\begin{array}{ll}{\text { a. } \mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}} & {\text { b. } \mathbf{u} \times \mathbf{v}=-(\mathbf{v} \times \mathbf{u})} \\ {\mathbf{c} \cdot(-\mathbf{u}) \times \mathbf{v}=-(\mathbf{u} \times \mathbf{v})} & {}\end{array}\) d. \((c \mathbf{u}) \cdot \mathbf{v}=\mathbf{u} \cdot(c \mathbf{v})=c(\mathbf{u} \cdot \mathbf{v}) \quad\) (any number \(c )\) e. \(c(\mathbf{u} \times \mathbf{v})=(c \mathbf{u}) \times \mathbf{v}=\mathbf{u} \times(c \mathbf{v}) \quad\) (any number \(c )\) \(\mathbf{f} . \mathbf{u} \cdot \mathbf{u}=|\mathbf{u}|^{2} \quad\) g. \((\mathbf{u} \times \mathbf{u}) \cdot \mathbf{u}=0\) \(\mathbf{h} .(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{u}=\mathbf{v} \cdot(\mathbf{u} \times \mathbf{v})\)

Find the point of intersection of the lines \(x=t, y=\) $$-t+2, z=t+1,\( and \)x=2 s+2, y=s+3, z=5 s+6$$ and then find the plane determined by these lines.

In Exercises \(19-28\) , describe the given set with a single equation or with a pair of equations. The set of points in space that lie 2 units from the point \((0,0,1)\) and, at the same time, 2 units from the point \((0,0,-1)\)

In Exercises 25–30, express each vector as a product of its length and direction. $$ \frac{3}{5} \mathbf{i}+\frac{4}{5} \mathbf{k} $$

Sketch the surfaces in Exercises \(13-76\) $$ z=18-x^{2}-9 y^{2} $$

Given nonzero vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w},\) use dot product and cross product notation, as appropriate, to describe the following. a. The vector projection of \(\mathbf{u}\) onto \(\mathbf{v}\) b. A vector orthogonal to \(\mathbf{u}\) and \(\mathbf{v}\) c. A vector orthogonal to \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{w}\) d. The volume of the parallelepiped determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\)

In Exercises 29 and 30, find the plane determined by the intersecting lines. $$ \begin{array}{l}{\text { L1: } x=-1+t, \quad y=2+t, \quad z=1-t ; \quad-\infty

a. Cauchy-Schwartz inequality Use the fact that \(\mathbf{u} \cdot \mathbf{v}=\) \(|\mathbf{u}||\mathbf{v}| \cos \theta\) to show that the inequality \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) holds for any vectors \(\mathbf{u}\) and \(\mathbf{v} .\) b. Under what circumstances, if any, does \(|\mathbf{u} \cdot \mathbf{v}|\) equal \(|\mathbf{u}||\mathbf{v}| ?\) Give reasons for your answer.

Write inequalities to describe the sets in Exercises \(29-34\) The slab bounded by the planes \(z=0\) and \(z=1\) (planes included)

In Exercises 25–30, express each vector as a product of its length and direction. $$ \frac{1}{\sqrt{6}} \mathbf{i}-\frac{1}{\sqrt{6}} \mathbf{j}-\frac{1}{\sqrt{6}} \mathbf{k} $$

Given nonzero vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w},\) use dot product and cross product notation to describe the following. a. A vector orthogonal to \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{u} \times \mathbf{w}\) b. A vector orthogonal to \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}-\mathbf{v}\) c. A vector of length \(|\mathbf{u}|\) in the direction of \(\mathbf{v}\) d. The area of the parallelogram determined by \(\mathbf{u}\) and \(\mathbf{w}\)

In Exercises 29 and 30, find the plane determined by the intersecting lines. $$ \begin{array}{ll}{L 1 : \quad x=t,} & {y=3-3 t, \quad z=-2-t ; \quad-\infty