Chapter 10

\(29-32\) Find the scalar and vector projections of b onto a. $$\mathbf{a}=\mathbf{i}+\mathbf{j}+\mathbf{k}, \quad \mathbf{b}=\mathbf{i}-\mathbf{j}+\mathbf{k}$$

Find the tangential and normal components of the acceleration vector. $$\mathbf{r}(t)=\left(3 t-t^{3}\right) \mathbf{i}+3 t^{2} \mathbf{j}$$

Plot the space curve and its curvature function \(\kappa(t)\). Comment on how the curvature reflects the shape of the curve. $$\mathbf{r}(t)=\left\langle t e^{t}, e^{-t}, \sqrt{2} t\right\rangle, \quad-5 \leqslant t \leqslant 5$$

\(33-38=\) (a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$\mathbf{r}(t)=\left\langle t-2, t^{2}+1\right\rangle, \quad t=-1$$

Find the point at which the line \(x=3-t, y=2+t\) \(z=5 t\) intersects the plane \(x-y+2 z=9\)

Graph the surfaces \(z=x^{2}+y^{2}\) and \(z=1-y^{2}\) on a common screen using the domain \(|x| \leqslant 1.2,|y| \leqslant 1.2\) and observe the curve of intersection of these surfaces. Show that the projection of this curve onto the \(x y\) -plane is an ellipse.

Find the volume of the parallelepiped determined by the vectors a, \(\mathbf{b},\) and \(\mathbf{c} .\) $$\mathbf{a}=\langle 1,2,3\rangle, \quad \mathbf{b}=\langle- 1,1,2\rangle, \quad \mathbf{c}=\langle 2,1,4\rangle$$

Find the unit vectors that are parallel to the tangent line to the parabola \(y=x^{2}\) at the point \((2,4)\)

Show that the vector orth \(_{\mathrm{a}} \mathbf{b}=\mathbf{b}-\) proja \(_{\mathrm{a}} \mathbf{b}\) is orthogonal to a. (It is called an orthogonal projection of \(\mathbf{b} . )\)

If a particle with mass \(m\) moves with position vector \(\mathbf{r}(t),\) then its angular momentum is defined as \(\mathbf{L}(t)=m \mathbf{r}(t) \times \mathbf{v}(t)\) and its torque as \(\boldsymbol{\tau}(t)=m \mathbf{r}(t) \times \mathbf{a}(t)\). Show that \(\mathbf{L}^{\prime}(t)=\boldsymbol{\tau}(t) .\) Deduce that if \(\boldsymbol{\tau}(t)=\mathbf{0}\) for all \(t\) then \(\mathbf{L}(t)\) is constant. (This is the law of conservation of angular momentum.)

\(33-38=\) (a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=\left\langle t^{2}, t^{3}\right\rangle, \quad t=1 $$

Where does the line through \((1,0,1)\) and \((4,-2,2)\) intersect the plane \(x+y+z=6 ?\)

Show that the curve of intersection of the surfaces \(x^{2}+2 y^{2}-z^{2}+3 x=1\) and \(2 x^{2}+4 y^{2}-2 z^{2}-5 y=0\) lies in a plane.

Find the volume of the parallelepiped determined by the vectors a, \(\mathbf{b},\) and \(\mathbf{c} .\) $$\mathbf{a}=\mathbf{i}+\mathbf{j}, \quad \mathbf{b}=\mathbf{j}+\mathbf{k}, \quad \mathbf{c}=\mathbf{i}+\mathbf{j}+\mathbf{k}$$

(a) Find the unit vectors that are parallel to the tangent line to the curve \(y=2 \sin x\) at the point \((\pi / 6,1)\) . (b) Find the unit vectors that are perpendicular to the tangent line. (c) Sketch the curve \(y=2 \sin x\) and the vectors in parts (a) and (b), all starting at \((\pi / 6,1)\) .

The position function of a spaceship is $$\mathbf{r}(t)=(3+t) \mathbf{i}+(2+\ln t) \mathbf{j}+\left(7-\frac{4}{t^{2}+1}\right) \mathbf{k}$$ and the coordinates of a space station are \((6,4,9) .\) The captain wants the spaceship to coast into the space station. When should the engines be turned off?

\(33-38=\) (a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=\sin t \mathbf{i}+2 \cos t \mathbf{j}, \quad t=\pi / 4 $$

\(35-38=\) Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. $$x+y+z=1, \quad x-y+z=1$$

Find the volume of the parallelepiped with adjacent edges \(P Q, P R,\) and \(P S .\) $$P(-2,1,0), \quad Q(2,3,2), \quad R(1,4,-1), \quad S(3,6,1)$$

Find an equation of the set of all points equidistant from the points \(A(-1,5,3)\) and \(B(6,2,-2)\) . Describe the set.

(a) Draw the vectors a \(=\langle 3,2\rangle, \mathbf{b}=\langle 2,-1\rangle,\) and \(\mathbf{c}=\langle 7,1\rangle .\) (b) Show, by means of a sketch, that there are scalars \(s\) and tsuch that \(\mathbf{c}=s \mathbf{a}+t \mathbf{b} .\) (c) Use the sketch to estimate the values of \(s\) and \(t\) (d) Find the exact values of \(s\) and \(t .\)

If \(\mathbf{a}=\langle 3,0,-1\rangle,\) find a vector \(\mathbf{b}\) such that comp \(_{\mathbf{a}} \mathbf{b}=2\)

A rocket burning its onboard fuel while moving through space has velocity \(\mathbf{v}(t)\) and mass \(m(t)\) at time \(t .\) If the exhaust gases escape with velocity \(\mathbf{v}_{e}\) relative to the rocket, it can be deduced from Newton's Second Law of Motion that $$m \frac{d \mathbf{v}}{d t}=\frac{d m}{d t} \mathbf{v}_{e}$$ (a) Show that \(\mathbf{v}(t)=\mathbf{v}(0)-\ln \frac{m(0)}{m(t)} \mathbf{v}_{e}\) (b) For the rocket to accelerate in a straight line from rest to twice the speed of its own exhaust gases, what fraction of its initial mass would the rocket have to burn as fuel?

\(33-38=\) (a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j}, \quad t=0 $$

\(35-38=\) Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. $$2 x-3 y+4 z=5, \quad x+6 y+4 z=3$$

Find the volume of the parallelepiped with adjacent edges \(P Q, P R,\) and \(P S .\) $$P(3,0,1), \quad Q(-1,2,5), \quad R(5,1,-1), \quad S(0,4,2)$$

Find the volume of the solid that lies inside both of the spheres $$x^{2}+y^{2}+z^{2}+4 x-2 y+4 z+5=0$$ and $$x^{2}+y^{2}+z^{2}=4$$

Suppose that a and b are nonzero vectors. (a) Under what circumstances is \(\operatorname{comp}_{\mathbf{a}} \mathbf{b}=\operatorname{comp}_{\mathbf{b}} \mathbf{a} ?\) (b) Under what circumstances is \(\operatorname{proj}_{a} b=\operatorname{proj}_{b} a ?\)

Suppose that a and b are nonzero vectors that are not paral- lel and \(c\) is any vector in the plane determined by a and b. Give a geometric argument to show that \(\mathbf{c}\) can be written as \(\mathbf{c}=s \mathbf{a}+t \mathbf{b}\) for suitable scalars \(s\) and \(t .\) Then give an argument using components.

\(33-38=\) (a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=e^{2 t} \mathbf{i}+e^{t} \mathbf{j}, \quad t=0 $$

Use the scalar triple product to verify that the vectors \(\mathbf{u}=\mathbf{i}+5 \mathbf{j}-2 \mathbf{k}, \mathbf{v}=3 \mathbf{i}-\mathbf{j},\) and \(\mathbf{w}=5 \mathbf{i}+9 \mathbf{j}-4 \mathbf{k}\) are coplanar.

Find the distance between the spheres \(x^{2}+y^{2}+z^{2}=4\) and \(x^{2}+y^{2}+z^{2}=4 x+4 y+4 z-11.\)

If \(\mathbf{r}=\langle x, y, z\rangle\) and \(\mathbf{r}_{0}=\left\langle x_{0}, y_{0}, z_{0}\right\rangle,\) describe the set of all points \((x, y, z)\) such that \(\left|\mathbf{r}-\mathbf{r}_{0}\right|=1\)

Find the work done by a force \(\mathbf{F}=8 \mathbf{i}-6 \mathbf{j}+9 \mathbf{k}\) that moves an object from the point \((0,10,8)\) to the point \((6,12,20)\) along a straight line. The distance is measured in meters and the force in newtons.

\(35-38=\) Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. $$x+2 y+2 z=1, \quad 2 x-y+2 z=1$$

\(33-38=\) (a) Sketch the plane curve with the given vector equation. (b) Find \(\mathbf{r}^{\prime}(t) .\) (c) Sketch the position vector \(\mathbf{r}(t)\) and the tangent vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\) . $$ \mathbf{r}(t)=(1+\cos t) \mathbf{i}+(2+\sin t) \mathbf{j}, \quad t=\pi / 6 $$

Use the scalar triple product to determine whether the points \(A(1,3,2), B(3,-1,6), C(5,2,0),\) and \(D(3,6,-4)\) lie in the same plane.

Describe and sketch a solid with the following properties. When illuminated by rays parallel to the \(z\)-axis, its shadow is a circular disk. If the rays are parallel to the \(y\) -axis, its shadow is a square. If the rays are parallel to the \(x\)-axis, its shadow is an isosceles triangle.

If \(\mathbf{r}=\langle x, y\rangle, \mathbf{r}_{1}=\left\langle x_{1}, y_{1}\right\rangle,\) and \(\mathbf{r}_{2}=\left\langle x_{2}, y_{2}\right\rangle,\) describe the set of all points \((x, y)\) such that \(\left|\mathbf{r}-\mathbf{r}_{1}\right|+\left|\mathbf{r}-\mathbf{r}_{2}\right|=k\) where \(k>\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|\)

A tow truck drags a stalled car along a road. The chain makes an angle of \(30^{\circ}\) with the road and the tension in the chain is 1500 \(\mathrm{N} .\) How much work is done by the truck in pulling the car 1 \(\mathrm{km}\) ?

Find the vectors \(\mathbf{T}, \mathbf{N},\) and \(\mathbf{B}\) at the given point. $$\mathbf{r}(t)=\left\langle t^{2}, \frac{2}{3} t^{3}, t\right\rangle, \quad\left(1, \frac{2}{3}, 1\right)$$

\(39-44=\) Find the derivative of the vector function. $$ \mathbf{r}(t)=\left\langle t \sin t, t^{2}, t \cos 2 t\right\rangle $$

(a) Find parametric equations for the line of intersection of the planes \(x+y+z=1\) and \(x+2 y+2 z=1\) (b) Find the angle between these planes.

A sled is pulled along a level path through snow by a rope. A 30 -lb force acting at an angle of \(40^{\circ}\) above the horizontal moves the sled 80 ft. Find the work done by the force.

\(39-44=\) Find the derivative of the vector function. $$ \mathbf{r}(t)=\left\langle\tan t, \sec t, 1 / t^{2}\right\rangle $$

Find an equation of the plane with \(x\) -intercept \(a, y\) -intercept \(b,\) and \(z\) -intercept \(c .\)

A boat sails south with the help of a wind blowing in the direction \(S36^{\circ} \mathrm{E}\) with magnitude 400 \(\mathrm{lb}\) . Find the work done by the wind as the boat moves 120 \(\mathrm{ft.}\)

Find the vectors \(\mathbf{T}, \mathbf{N},\) and \(\mathbf{B}\) at the given point. $$\mathbf{r}(t)=\langle\cos t, \sin t, \ln \cos t\rangle, \quad(1,0,0)$$

Find equations of the normal plane and osculating plane of the curve at the given point. $$x=2 \sin 3 t, y=t, z=2 \cos 3 t ; \quad(0, \pi,-2)$$

\(39-44=\) Find the derivative of the vector function. $$ \mathbf{r}(t)=e^{t^{2}} \mathbf{i}-\mathbf{j}+\ln (1+3 t) \mathbf{k} $$