Chapter 10

Find equations of the planes that are parallel to the plane \(x+2 y-2 z=1\) and two units away from it.

Show that the lines with symmetric equations \(x=y=z\) and \(x+1=y / 2=z / 3\) are skew, and find the distance between these lines. [Hint: The skew lines lie in parallel planes.

Find parametric equations for the tangent line to the curve \(x=t \cos t, y=t, z=t \sin t\) at the point \((-\pi, \pi, 0) .\) Illustrate by graphing both the curve and the tangent line on a common screen.

Find the distance between the skew lines with parametric equations \(x=1+t, y=1+6 t, z=2 t,\) and \(x=1+2 s\) \(y=5+15 s, z=-2+6 s\)

(a) Find the point of intersection of the tangent lines to the curve \(\mathbf{r}(t)=\langle\sin \pi t, 2 \sin \pi t, \cos \pi t\rangle\) at the points where \( t=0\) and \(t=0.5\) . (b) Illustrate by graphing the curve and both tangent lines.

Let \(L_{1}\) be the line through the origin and the point \((2,0,-1)\) Let \(L_{2}\) be the line through the points \((1,-1,1)\) and \((4,1,3)\) ) Find the distance between \(L_{1}\) and \(L_{2} .\)

The curves \(\mathbf{r}_{1}(t)=\left\langle t, t^{2}, t^{3}\right\rangle\) and \(\mathbf{r}_{2}(t)=\langle\sin t, \sin 2 t, t\rangle\) intersect at the origin. Find their angle of intersection correct to the nearest degree.

Let \(L_{1}\) be the line through the points \((1,2,6)\) and \((2,4,8) .\) Let \(L_{2}\) be the line of intersection of the planes \(\pi_{1}\) and \(\pi_{2}\) where \(\pi_{1}\) is the plane \(x-y+2 z+1=0\) and \(\pi_{2}\) is the plane through the points \((3,2,-1),(0,0,1),\) and \((1,2,1)\) . Calculate the distance between \(L_{1}\) and \(L_{2}\) .

At what point do the curves \(\mathbf{r}_{1}(t)=\left\langle t, 1-t, 3+t^{2}\right\rangle\) and \(\mathbf{r}_{2}(s)=\left\langle 3-s, s-2, s^{2}\right\rangle\) intersect? Find their angle of intersection correct to the nearest degree.

\(59-64=\) Evaluate the integral. $$ \int_{0}^{2}\left(t \mathbf{i}-t^{3} \mathbf{j}+3 t^{5} \mathbf{k}\right) d t $$

If \(a, b,\) and \(c\) are not all \(0,\) show that the equation \(a x+b y+c z+d=0\) represents a plane and \(\langle a, b, c\rangle\) is a normal vector to the plane. Hint: Suppose \( a \neq 0\) and rewrite the equation in the form $$a\left(x+\frac{d}{a}\right)+b(y-0)+c(z-0)=0$$

When \(a \ne 0\), there are two solutions to \\(ax^2 + bx + c = 0\\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$ $$ \int_{0}^{1}\left(\frac{4}{1+t^{2}} \mathbf{j}+\frac{2 t}{1+t^{2}} \mathbf{k}\right) d t $$

Give a geometric description of each family of planes. $$ \begin{array}{l}{\text { (a) } x+y+z=c \quad \text { (b) } x+y+c z=1} \\\ {\text { (c) } y \cos \theta+z \sin \theta=1}\end{array} $$

When \(a \ne 0\), there are two solutions to \\(ax^2 + bx + c = 0\\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$ $$ \int_{0}^{\pi / 2}\left(3 \sin ^{2} t \cos t \mathbf{i}+3 \sin t \cos ^{2} t \mathbf{j}+2 \sin t \cos t \mathbf{k}\right) d t $$

When \(a \ne 0\), there are two solutions to \\(ax^2 + bx + c = 0\\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$ $$ \int_{1}^{2}\left(t^{2} \mathbf{i}+t \sqrt{t-1} \mathbf{j}+t \sin \pi t \mathbf{k}\right) d t $$

When \(a \ne 0\), there are two solutions to \\(ax^2 + bx + c = 0\\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$ $$ \int\left(\sec ^{2} t \mathbf{i}+t\left(t^{2}+1\right)^{3} \mathbf{j}+t^{2} \ln t \mathbf{k}\right) d t $$

Find \(\mathbf{r}(t)\) if \(\mathbf{r}^{\prime}(t)=2 t \mathbf{i}+3 t^{2} \mathbf{j}+\sqrt{t} \mathbf{k}\) and \(\mathbf{r}(1)=\mathbf{i}+\mathbf{j}\)

Find \(\mathbf{r}(t)\) if \(\mathbf{r}^{\prime}(t)=t \mathbf{i}+e^{t} \mathbf{j}+t e^{t} \mathbf{k}\) and \(\mathbf{r}(0)=\mathbf{i}+\mathbf{j}+\mathbf{k}\)

If two objects travel through space along two different curves, it's often important to know whether they will col- lide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions $$\mathbf{r}_{1}(t)=\left\langle t^{2}, 7 t-12, t^{2}\right\rangle \quad \mathbf{r}_{2}(t)=\left\langle 4 t-3, t^{2}, 5 t-6\right\rangle$$ for \(t \geqslant 0 .\) Do the particles collide?

Two particles travel along the space curves $$\mathbf{r}_{1}(t)=\left\langle t, t^{2}, t^{3}\right\rangle \quad \mathbf{r}_{2}(t)=\langle 1+2 t, 1+6 t, 1+14 t\rangle$$ Do the particles collide? Do their paths intersect?

Suppose \(\mathbf{u}\) and \(\mathbf{v}\) are vector functions that possess limits as \(t \rightarrow a\) and let \(c\) be a constant. Prove the following properties of limits. (a) $$\lim _{t \rightarrow a}[\mathbf{u}(t)+\mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t)+\lim _{t \rightarrow a} \mathbf{v}(t)$$ (b) $$\lim _{r \rightarrow a} c \mathbf{u}(t)=c \lim _{t \rightarrow a} \mathbf{u}(t)$$ (c) $$\lim _{t \rightarrow a}[\mathbf{u}(t) \cdot \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \cdot \lim _{t \rightarrow a} \mathbf{v}(t)$$ (d) $$\lim _{t \rightarrow a}[\mathbf{u}(t) \times \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \times \lim _{t \rightarrow a} \mathbf{v}(t)$$

Show that \(\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{b}\) if and only if for every \(\varepsilon>0\) there is a number \(\delta>0\) such that \(|\mathbf{r}(t)-\mathbf{b}|<\varepsilon\) whenever \(0<|t-a|<\delta\)

Find \(f^{\prime}(2),\) where \(f(t)=\mathbf{u}(t) \cdot \mathbf{v}(t), \mathbf{u}(2)=\langle 1,2,-1\rangle\) $$\mathbf{u}^{\prime}(2)=\langle 3,0,4\rangle,\( and \)\mathbf{v}(t)=\left\langle t, t^{2}, t^{3}\right\rangle$$

Show that if \(\mathbf{r}\) is a vector function such that \(\mathbf{r}^{\prime \prime}\) cxists, then $$\frac{d}{d t}\left[\mathbf{r}(t) \times \mathbf{r}^{\prime}(t)\right]=\mathbf{r}(t) \times \mathbf{r}^{\prime \prime}(t)$$

Find an expression for $$\frac{d}{d t}[\mathbf{u}(t) \cdot(\mathbf{v}(t) \times \mathbf{w}(t))]$$

If \(\mathbf{r}(t) \neq \mathbf{0},$$ show that \)\frac{d}{d t}|\mathbf{r}(t)|=\frac{1}{|\mathbf{r}(t)|} \mathbf{r}(t) \cdot \mathbf{r}^{\prime}(t)$$

If a curve has the property that the position vector \(\mathbf{r}(t)\) is always perpendicular to the tangent vector \(\mathbf{r}^{\prime}(t),\) show that the curve lies on a sphere with center the origin.

If $$\mathbf{u}(t)=\mathbf{r}(t) \cdot\left[\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)\right],$$ show that $$ \mathbf{u}^{\prime}(t)=\mathbf{r}(t) \cdot\left[\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime \prime}(t)\right] $$