Chapter 11

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(0,0,0), \quad P_{2}(2,-2,-2)$$

Compute \((\mathbf{i} \times \mathbf{j}) \times \mathbf{j}\) and \(\mathbf{i} \times(\mathbf{j} \times \mathbf{j}) .\) What can you conclude about the associativity of the cross product?

Sketch the surfaces HYPERBOLOIDS $$\left(y^{2} / 4\right)-\left(x^{2} / 4\right)-z^{2}=1$$

Cancelation in dot products In real-number multiplication, if \(u v_{1}=u v_{2}\) and \(u \neq 0,\) we can cancel the \(u\) and conclude that \(v_{1}=v_{2} .\) Does the same rule hold for the dot product? That is, if \(\mathbf{u} \cdot \mathbf{v}_{1}=\mathbf{u} \cdot \mathbf{v}_{2}\) and \(\mathbf{u} \neq \mathbf{0},\) can you conclude that \(\mathbf{v}_{1}=\mathbf{v}_{2} ?\) Give reasons for your answer.

Find the plane containing the intersecting lines. \(\begin{array}{ll}L 1: x=-1+t, & y=2+t, z=1-t ; \quad-\infty

Express each vector as a product of its length and direction. $$\frac{\mathbf{i}}{\sqrt{3}}+\frac{\mathbf{j}}{\sqrt{3}}+\frac{\mathbf{k}}{\sqrt{3}}$$

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(5,3,-2), \quad P_{2}(0,0,0)$$

Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be vectors. Which of the following make sense, and which do not? Give reasons for your answers. a. \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\) b. \(\mathbf{u} \times(\mathbf{v} \cdot \mathbf{w})\) c. \(\mathbf{u} \times(\mathbf{v} \times \mathbf{w})\) d. \(\mathbf{u} \cdot(\mathbf{v} \cdot \mathbf{w})\)

Sketch the surfaces HYPERBOLIC PARABOLOIDS $$y^{2}-x^{2}=z$$

If \(u\) and \(v\) are orthogonal, show that proje \(\mathbf{u}=0\)

Find a plane through \(P_{0}(2,1,-1)\) and perpendicular to the line of intersection of the planes \(2 x+y-z=3, x+2 y+z=2\)

Find the vectors whose lengths and directions are given. Try to do the calculations without writing. $$\begin{aligned} &\begin{array}{ll} \text { Length } & \text { Direction } \\ \hline \text { a. } 2 & \text { i } \\ \text { b. } \sqrt{3} & -\text { k } \\ \text { c. } \frac{1}{2} & \frac{3}{5} \mathbf{j}+\frac{4}{5} \mathbf{k} \\ \text { d. } 7 & \frac{6}{7} \mathbf{i}-\frac{2}{7} \mathbf{j}+\frac{3}{7} \mathbf{k} \end{array}\\\ \end{aligned}$$

Find the distance from the point (3,-4,2) to the a. \(x y\) -plane b. yz-plane c. \(x z\) -plane

Show that except in degenerate cases, \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w}\) lies in the plane of \(\mathbf{u}\) and \(\mathbf{v},\) whereas \(\mathbf{u} \times(\mathbf{v} \times \mathbf{w})\) lies in the plane of \(\mathbf{v}\) and \(\mathbf{w} .\) What are the degenerate cases?

Sketch the surfaces HYPERBOLIC PARABOLOIDS $$x^{2}-y^{2}=z$$

A force \(\mathbf{F}=2 \mathbf{i}+\mathbf{j}-3 \mathbf{k}\) is applied to a spacecraft with velocity vector \(\mathbf{v}=3 \mathbf{i}-\mathbf{j} .\) Express \(\mathbf{F}\) as a sum of a vector parallel to \(\mathbf{v}\) and a vector orthogonal to \(\mathbf{v}\).

Find a plane through the points \(P_{1}(1,2,3), P_{2}(3,2,1)\) and perpendicular to the plane \(4 x-y+2 z=7\)

Find the vectors whose lengths and directions are given. Try to do the calculations without writing. $$\begin{array}{ll} \text { Length } & \text { Direction } \\ \hline \text { a. } 7 & -\mathbf{j} \\ \text { b. } \sqrt{2} & -\frac{3}{5} \mathbf{i}-\frac{4}{5} \mathbf{k} \\ \text { c. } \frac{13}{12} & \frac{3}{13} \mathbf{i}-\frac{4}{13} \mathbf{j}-\frac{12}{13} \mathbf{k} \\ \text { d. } a>0 & \frac{1}{\sqrt{2}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}-\frac{1}{\sqrt{6}} \mathbf{k} \end{array}$$

Find the distance from the point (-2,1,4) to the a. plane \(x=3\) b. plane \(y=-5\) c. plane \(z=-1\)

If \(\mathbf{u} \times \mathbf{v}=\mathbf{u} \times \mathbf{w}\) and \(\mathbf{u} \neq \mathbf{0}\) then does \(\mathbf{v}=\mathbf{w} ?\) Give reasons for your answer.

Sketch the surfaces ASSORTED $$z=1+y^{2}-x^{2}$$

Show that \(\mathbf{v}=a \mathbf{i}+b \mathbf{j}\) is perpendicular to the line \(a x+b y=c\) (Hint: For \(a\) and \(b\) nonzero, establish that the slope of the vector \(v\) is the negative reciprocal of the slope of the given line. Also verify the statement when \(a=0\) or \(\bar{b}=0\)

find the distance from the point to the line. $$(0,0,12) ; \quad x=4 t, \quad y=-2 t, \quad z=2 t$$

Find a vector of magnitude 7 in the direction of \(\mathbf{v}=12 \mathbf{i}-5 \mathbf{k}\).

Find the distance from the point (4,3,0) to the a. \(x\) -axis b. \(y\) -axis c. z-axis

If \(\mathbf{u} \neq \mathbf{0}\) and if \(\mathbf{u} \times \mathbf{v}=\mathbf{u} \times \mathbf{w}\) and \(\mathbf{u} \cdot \mathbf{v}=\mathbf{u} \cdot \mathbf{w},\) then does \(\mathbf{v}=\mathbf{w} ?\) Give reasons for your answer.

Sketch the surfaces ASSORTED $$4 x^{2}+4 y^{2}=z^{2}$$

Show that the vector \(\mathbf{v}=a \mathbf{i}+b \mathbf{j}\) is parallel to the line \(b x-a y=c\) (Hint: For \(a\) and \(b\) nonzero, establish that the slope of the line segment representing \(\mathbf{v}\) is the same as the slope of the given line. Also verify the statement when \(a=0\) or \(b=0\)

find the distance from the point to the line. $$(0,0,0) ; \quad x=5+3 t, \quad y=5+4 t, \quad z=-3-5 t$$

Find a vector of magnitude 3 in the direction opposite to the direction of \(\mathbf{v}=(1 / 2) \mathbf{i}-(1 / 2) \mathbf{j}-(1 / 2) \mathbf{k}\).

Find the distance from the a. \(x\) -axis to the plane \(z=3\) b. origin to the plane \(2=z-x\) c. point (0,4,0) to the plane \(y=x\)

Find the areas of the parallelograms whose vertices are given. $$A(1,0), \quad B(0,1), \quad C(-1,0), \quad D(0,-1)$$

Sketch the surfaces ASSORTED $$y=-\left(x^{2}+z^{2}\right)$$

Find a. the direction of \(\vec{P}_{1} \vec{P}_{2}\) and \(\mathbf{b}\). the midpoint of line segment \(P_{1} P_{2}\). $$P_{1}(-1,1,5) \quad P_{2}(2,5,0)$$

find the distance from the point to the line. $$(2,1,3) ; \quad x=2+2 t, \quad y=1+6 t, \quad z=3$$

Describe the given set with a single equation or with a pair of equations. The plane perpendicular to the a. \(x\) -axis at (3,0,0) b. \(y\) -axis at (0,-1,0) c. \(z\) -axis at (0,0,-2)

Find the areas of the parallelograms whose vertices are given. $$A(0,0), \quad B(7,3), \quad C(9,8), \quad D(2,5)$$

Sketch the surfaces ASSORTED $$16 x^{2}+4 y^{2}=1$$

Find a. the direction of \(\vec{P}_{1} \vec{P}_{2}\) and \(\mathbf{b}\). the midpoint of line segment \(P_{1} P_{2}\). $$P_{1}(1,4,5) \quad P_{2}(4,-2,7)$$

find the distance from the point to the line. $$(2,1,-1) ; \quad x=2 t, \quad y=1+2 t, \quad z=2 t$$

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 b. y-axis c. z-axis

Find the areas of the parallelograms whose vertices are given. $$A(-1,2), \quad B(2,0), \quad C(7,1), \quad D(4,3)$$

Sketch the surfaces ASSORTED $$x^{2}+y^{2}-z^{2}=4$$

Find a. the direction of \(\vec{P}_{1} \vec{P}_{2}\) and \(\mathbf{b}\). the midpoint of line segment \(P_{1} P_{2}\). $$P_{1}(3,4,5) \quad P_{2}(2,3,4)$$

find the distance from the point to the line. $$(3,-1,4) ; \quad x=4-t, \quad y=3+2 t, \quad z=-5+3 t$$

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 b. yz-plane c. \(x z\) -plane

Find the areas of the parallelograms whose vertices are given. $$A(-6,0), \quad B(1,-4), \quad C(3,1), \quad D(-4,5)$$

Sketch the surfaces ASSORTED $$x^{2}+z^{2}=y$$

Find a. the direction of \(\vec{P}_{1} \vec{P}_{2}\) and \(\mathbf{b}\). the midpoint of line segment \(P_{1} P_{2}\). $$P_{1}(0,0,0) \quad P_{2}(2,-2,-2)$$

find the distance from the point to the line. $$(-1,4,3) ; \quad x=10+4 t, \quad y=-3, \quad z=4 t$$