Chapter 10

A wrench 30 \(\mathrm{cm}\) long lies along the positive \(y\) -axis and grips a bolt at the origin. A force is applied in the direction \(\langle 0,3,-4\rangle\) at the end of the wrench. Find the magnitude of the force needed to supply 100 \(\mathrm{N} \cdot \mathrm{m}\) of torque to the bolt.

Use a scalar projection to show that the distance from a point \(P_{1}\left(x_{1}, y_{1}\right)\) to the line \(a x+b y+c=0\) is $$\frac{\left|a x_{1}+b y_{1}+c\right|}{\sqrt{a^{2}+b^{2}}}$$ Use this formula to find the distance from the point \((-2,3)\) to the line \(3 x-4 y+5=0\)

Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Find equations of the normal plane and osculating plane of the curve at the given point. $$x=t, y=t^{2}, z=t^{3} ; \quad(1,1,1)$$

(a) Find the point at which the given lines intersect: $$\begin{aligned} \mathbf{r} &=\langle 1,1,0\rangle+ t\langle 1,-1,2\rangle \\\ \mathbf{r} &=\langle 2,0,2\rangle+ s\langle- 1,1,0\rangle \end{aligned}$$ (b) Find an equation of the plane that contains these lines.

\(39-44=\) Find the derivative of the vector function. $$ \mathbf{r}(t)=a t \cos 3 t \mathbf{i}+b \sin ^{3} t \mathbf{j}+c \cos ^{3} t \mathbf{k} $$

Let \(\mathbf{v}=5 \mathbf{j}\) and let \(\mathbf{u}\) be a vector with length 3 that starts at the origin and rotates in the \(x y\) -plane. Find the maximum and minimum values of the length of the vector \(\mathbf{u} \times \mathbf{v} .\) In what direction does \(\mathbf{u} \times \mathbf{v}\) point?

If \(\mathbf{r}=\langle x, y, z\rangle, \mathbf{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle,\) and \(\mathbf{b}=\left\langle b_{1}, b_{2}, b_{3}\right\rangle\) show that the vector equation \((\mathbf{r}-\mathbf{a}) \cdot(\mathbf{r}-\mathbf{b})=0\) represents a sphere, and find its center and radius.

Find equations of the osculating circles of the ellipse \(9 x^{2}+4 y^{2}=36\) at the points \((2,0)\) and \((0,3) .\) Use a graphing calculator or computer to graph the ellipse and both osculating circles on the same screen.

Find parametric equations for the line through the point \((0,1,2)\) that is parallel to the plane \(x+y+z=2\) and perpendicular to the line \(x=1+t, y=1-t, z=2 t\)

\(39-44=\) Find the derivative of the vector function. $$ ^{3} t \mathbf{k} $$

If \(\mathbf{a} \cdot \mathbf{b}=\sqrt{3}\) and \(\mathbf{a} \times \mathbf{b}=\langle 1,2,2\rangle,\) find the angle between a and \(\mathbf{b} .\)

Find the angle between a diagonal of a cube and one of its edges.

Find parametric equations for the line through the point \((0,1,2)\) that is perpendicular to the line \(x=1+t\) \(y=1-t, z=2 t\) and intersects this line.

(a) Find all vectors \(\mathbf{v}\) such that $$\langle 1,2,1\rangle \times \mathbf{v}=\langle 3,1,-5\rangle$$ (b) Explain why there is no vector \(\mathbf{v}\) such that $$\langle 1,2,1\rangle \times \mathbf{v}=\langle 3,1,5\rangle$$

\(39-44=\) Find the derivative of the vector function. $$ \mathbf{r}(t)=t \mathbf{a} \times(\mathbf{b}+t \mathbf{c}) $$

Find the angle between a diagonal of a cube and a diagonal of one of its faces.

At what point on the curve \(x=t^{3}, y=3 t, z=t^{4}\) is the normal plane parallel to the plane \(6 x+6 y-8 z=1 ?\)

Which of the following four planes are parallel? Are any of them identical? $$\begin{array}{ll}{P_{1} : 3 x+6 y-3 z=6} & {P_{2} : 4 x-12 y+8 z=5} \\\ {P_{3} : 9 y=1+3 x+6 z} & {P_{4} : z=x+2 y-2}\end{array}$$

\(45-46\) . Find the unit tangent vector \(\mathbf{T}(t)\) at the point with the given value of the parameter \(t\) . $$\mathbf{r}(t)=\cos t \mathbf{i}+3 t \mathbf{j}+2 \sin 2 t \mathbf{k}, \quad t=0$$

A molecule of methane, \(\mathrm{CH}_{4},\) is structured with the four hydrogen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by the \(\mathrm{H}-\mathrm{C}-\mathrm{H}\) combination; it is the angle between the lines that join the carbon atom to two of the hydrogen atoms. Show that the bond angle is about \(109.5^{\circ}\) Hint: Take the vertices of the tetrahedron to be the points \((1,0,0),(0,1,0),(0,0,1),\) and \((1,1,1),\) as shown in the figure. Then the centroid is \(\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) . ]\)

(a) Let \(P\) be a point not on the plane that passes through the points \(Q, R,\) and \(S .\) Show that the distance \(d\) from \(P\) to the plane is $$\begin{aligned} d &=\frac{|\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})|}{|\mathbf{a} \times \mathbf{b}|} \\ \text { where a }=\vec{Q R}, \mathbf{b} &=\vec{Q S}, \text { and } \mathbf{c}=\vec{Q P} .\end{aligned}$$ (b) Use the formula in part (a) to find the distance from the point \(P(2,1,4)\) to the plane through the points \(Q(1,0,0),\) \(R(0,2,0),\) and \(S(0,0,3).\)

Which of the following four lines are parallel? Are any of them identical? $$L_{1} : x=1+6 t, \quad y=1-3 t, \quad z=12 t+5\( \)L_{2} : x=1+2 t, \quad y=t, \quad z=1+4 t\( \)L_{3} : 2 x-2=4-4 y=z+1\( \)L_{4} : \mathbf{r}=\langle 3,1,5\rangle+ t\langle 4,2,8\rangle$$

\(45-46\) . Find the unit tangent vector \(\mathbf{T}(t)\) at the point with the given value of the parameter \(t\) . $$\mathbf{r}(t)=\left\langle t^{3}+3 t, t^{2}+1,3 t+4\right\rangle, \quad t=1$$

If \(\mathbf{c}=|\mathbf{a}| \mathbf{b}+|\mathbf{b}| \mathbf{a},\) where \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are all nonzero vectors, show that \(\mathbf{c}\) bisects the angle between a and b.

Show that the curvature \(\kappa\) is related to the tangent and normal vectors by the equation $$\frac{d \mathbf{T}}{d s}=\kappa \mathbf{N}$$

Show that $$|\mathbf{a} \times \mathbf{b}|^{2}=|\mathbf{a}|^{2}|\mathbf{b}|^{2}-(\mathbf{a} \cdot \mathbf{b})^{2}.$$

If $$\mathbf{r}(t)=\left\langle t, t^{2}, t^{3}\right\rangle,\( find \)\mathbf{r}^{\prime}(t), \mathbf{T}(1), \mathbf{r}^{\prime \prime}(t),\( and \)\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)$$

Show that the curvature of a plane curve is \(\kappa=|d \phi / d s|,\) where \(\phi\) is the angle between \(T\) and \(\mathbf{i} ;\) that is, \(\phi\) is the angle of inclination of the tangent line.

If \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0},\) show that $$\mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{c}=\mathbf{c} \times \mathbf{a}$$

Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.

(a) Show that \(d \mathbf{B} / d s\) is perpendicular to \(\mathbf{B}\) . (b) Show that \(d \mathbf{B} / d s\) is perpendicular to \(\mathbf{T}\) . (c) Deduce from parts (a) and (b) that \(d \mathbf{B} / d s=-\tau(s) \mathbf{N}\) for some number \(\tau(s)\) called the torsion of the curve. (The torsion measures the degree of twisting of a curve.) (d) Show that for a planc curve the torsion is \(\tau(s)=0\) .

Prove that $$(\mathbf{a}-\mathbf{b}) \times(\mathbf{a}+\mathbf{b})=2(\mathbf{a} \times \mathbf{b})$$

\(49-50=\) Find the distance from the point to the given plane. $$(1,-2,4), \quad 3 x+2 y+6 z=5$$

\(49-52=\) Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x=1+2 \sqrt{t}, \quad y=t^{3}-t, \quad z=t^{3}+t ; \quad(3,0,2)$$

The following formulas, called the Frenet-Serret formulas, are of fundamental importance in differential geometry: $$\begin{array}{l}{\text { 1. } d \mathbf{T} / d s=\kappa \mathbf{N}} \\\ {\text { 2. } d \mathbf{N} / d s=-\kappa \mathbf{T}+\tau \mathbf{B}} \\\ {\text { 3. } d \mathbf{B} / d s=-\tau \mathbf{N}}\end{array}$$ (Formula 1 comes from Exercise 47 and Formula 3 comes from Exercise \(49 .\) ) Use the fact that \(\mathbf{N}=\mathbf{B} \times \mathbf{T}\) to deduce Formula 2 from Formulas 1 and 3 .

\(49-50=\) Find the distance from the point to the given plane. $$(-6,3,5), \quad x-2 y-4 z=8$$

The Triangle Inequality for vectors is $$|\mathbf{a}+\mathbf{b}| \leqslant|\mathbf{a}|+|\mathbf{b}|$$ (a) Give a geometric interpretation of the Triangle Inequality. (b) Use the Cauchy-Schwarz Inequality from Exercise 49 to prove the Triangle Inequality. [Hint: Use the fact that \(|\mathbf{a}+\mathbf{b}|^{2}=(\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}+\mathbf{b})\) and use Property 3 of the dot product.]

\(51-52=\) Find the distance between the given parallel planes. $$2 x-3 y+z=4, \quad 4 x-6 y+2 z=3$$

\(49-52=\) Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x=e^{-t} \cos t, \quad y=e^{-t} \sin t, \quad z=e^{-t_{ ;}},(1,0,1)$$

Prove that $$(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=\left| \begin{array}{ll}{\mathbf{a} \cdot \mathbf{c}} & {\mathbf{b} \cdot \mathbf{c}} \\\ {\mathbf{a} \cdot \mathbf{d}} & {\mathbf{b} \cdot \mathbf{d}}\end{array}\right|$$

\(51-52=\) Find the distance between the given parallel planes. $$6 z=4 y-2 x, \quad 9 z=1-3 x+6 y$$

\(49-52=\) Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x=\sqrt{t^{2}+3}, \quad y=\ln \left(t^{2}+3\right), \quad z=t ; \quad(2, \ln 4,1)$$

Show that if \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}-\mathbf{v}\) are orthogonal, then the vectors \(\mathbf{u}\) and \(\mathbf{v}\) must have the same length.

The DNA molecule has the shape of a double helix (see Figure 3 on page 582 ). The radius of each helix is about 10 angstroms \(\left(1 \mathrm{A}=10^{-8} \mathrm{cm}\right)\) . Each helix rises about 34 A during each complete turn, and there are about \(2.9 \times 10^{8}\) complete turns. Estimate the length of each helix.

Suppose that \(\mathbf{a} \neq \mathbf{0}\). $$\begin{array}{l}{\text { (a) If } \mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}, \text { does it follow that } \mathbf{b}=\mathbf{c} ?} \\\ {\text { (b) If } \mathbf{a} \times \mathbf{b}=\mathbf{a} \times \mathbf{c}, \text { does it follow that } \mathbf{b}=\mathbf{c} ?} \\ {\text { (c) If } \mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c} \text { and } \mathbf{a} \times \mathbf{b}=\mathbf{a} \times \mathbf{c}, \text { does it follow }} \\ {\text { that } \mathbf{b}=\mathbf{c} ?}\end{array}$$

Show that the distance between the parallel planes \(a x+b y+c z+d_{1}=0\) and \(a x+b y+c z+d_{2}=0\) is $$D=\frac{\left|d_{1}-d_{2}\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$

Find a vector equation for the tangent line to the curve of intersection of the cylinders \(x^{2}+y^{2}=25\) and \(y^{2}+z^{2}=20\) at the point \((3,4,2)\)

Let's consider the problem of designing a railroad track to make a smooth transition between sections of straight track. Existing track along the negative \(x\) -axis is to be joined smoothly to a track along the line \(y=1\) for \(x \geqslant 1\) (a) Find a polynomial \(P=P(x)\) of degree 5 such that the function \(F\) defined by $$F(x)=\left\\{\begin{array}{ll}{0} & {\text { if } x \leqslant 0} \\ {P(x)} & {\text { if } 0

If \(\mathbf{v}_{1}, \mathbf{v}_{2},\) and \(\mathbf{v}_{3}\) are noncoplanar vectors, let $$\begin{array}{c}{\mathbf{k}_{1}=\frac{\mathbf{v}_{2} \times \mathbf{v}_{3}}{\mathbf{v}_{1} \cdot\left(\mathbf{v}_{2} \times \mathbf{v}_{3}\right)} \quad \mathbf{k}_{2}=\frac{\mathbf{v}_{3} \times \mathbf{v}_{1}}{\mathbf{v}_{1} \cdot\left(\mathbf{v}_{2} \times \mathbf{v}_{3}\right)}} \\ {\mathbf{k}_{3}=\frac{\mathbf{v}_{1} \times \mathbf{v}_{2}}{\mathbf{v}_{1} \cdot\left(\mathbf{v}_{2} \times \mathbf{v}_{3}\right)}}\end{array}$$ (These vectors occur in the study of crystallography. Vectors of the form \(n_{1} \mathbf{v}_{1}+n_{2} \mathbf{v}_{2}+n_{3} \mathbf{v}_{3},\) where each \(n_{i}\) is an integer, form a lattice for a crystal. Vectors written similarly in terms of \(\mathbf{k}_{1}, \mathbf{k}_{2},\) and \(\mathbf{k}_{3}\) form the reciprocal lattice.) $$\begin{array}{l}{\text { (a) Show that } \mathbf{k}_{j} \text { is perpendicular to } \mathbf{v}_{j} \text { if } i \neq j}. \\ {\text { (b) Show that } \mathbf{k}_{i} \cdot \mathbf{v}_{i}=1 \text { for } i=1,2,3}. \\\ {\text { (c) Show that } \mathbf{k}_{1} \cdot\left(\mathbf{k}_{2} \times \mathbf{k}_{3}\right)=\frac{1}{\mathbf{v}_{1} \cdot\left(\mathbf{v}_{2} \times \mathbf{v}_{3}\right)}}.\end{array}$$