Chapter 11

Name and sketch the graph of each of the following equations in three-space. $$ 4 x^{2}+36 y^{2}=144 $$

In Problems 1-6, sketch the curve over the indicated domain for \(t\). Find \(\mathbf{v}, \mathbf{a}, \mathbf{T}\), and \(\kappa\) at the point where \(t=t_{1} .\) $$ \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j} ; \quad 0 \leq t \leq 2 ; t_{1}=1 $$

Find the parametric equations of the line through the given pair of points. \((1,-2,3),(4,5,6)\)

In Problems \(1-8\), find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow 1}\left[2 t \mathbf{i}-t^{2} \mathbf{j}\right] $$

Let \(\quad \mathbf{a}=-3 \mathbf{i}+2 \mathbf{j}-2 \mathbf{k}, \quad \mathbf{b}=-\mathbf{i}+2 \mathbf{j}-4 \mathbf{k}, \quad\) and \(\mathbf{c}=7 \mathbf{i}+3 \mathbf{j}-4 \mathbf{k} .\) Find each of the following: (a) \(\mathbf{a} \times \mathbf{b}\) (b) \(\mathbf{a} \times(\mathbf{b}+\mathbf{c})\) (c) \(\mathbf{a} \cdot(\mathbf{b}+\mathbf{c})\) (d) \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})\)

Let \(\mathbf{a}=-2 \mathbf{i}+3 \mathbf{j}, \mathbf{b}=2 \mathbf{i}-3 \mathbf{j}\), and \(\mathbf{c}=-5 \mathbf{j}\). Find each of the following: (a) \(2 \mathbf{a}-4 \mathbf{b}\) (b) \(\mathbf{a} \cdot \mathbf{b}\) (c) \(\mathbf{a} \cdot(\mathbf{b}+\mathbf{c})\) (d) \((-2 \mathbf{a}+3 \mathbf{b}) \cdot 5 \mathbf{c}\) (e) \(\|\mathbf{a}\| \mathbf{c} \cdot \mathbf{a}\) (f) \(\mathbf{b} \cdot \mathbf{b}-\|\mathbf{b}\|\)

Name and sketch the graph of each of the following equations in three-space. $$ y^{2}+z^{2}=15 $$

Change the following from cylindrical to spherical coordinates. (a) \((1, \pi / 2,1)\) (b) \((-2, \pi / 4,2)\)

sketch the curve over the indicated domain for \(t\). Find \(\mathbf{v}, \mathbf{a}, \mathbf{T}\), and \(\kappa\) at the point where \(t=t_{1} .\) $$ \mathbf{r}(t)=t^{2} \mathbf{i}+(2 t+1) \mathbf{j} ; \quad 0 \leq t \leq 2 ; t_{1}=1 $$

Find the parametric equations of the line through the given pair of points. \((2,-1,-5),(7,-2,3)\)

Find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow 3}\left[2(t-3)^{2} \mathbf{i}-7 t^{3} \mathbf{j}\right] $$

If \(\mathbf{a}=\langle 3,3,1\rangle, \mathbf{b}=\langle-2,-1,0\rangle\), and \(\mathbf{c}=\langle-2,-3,-1\rangle\), find each of the following: (a) \(\mathbf{a} \times \mathbf{b}\) (b) \(\mathbf{a} \times(\mathbf{b}+\mathbf{c})\) (c) \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})\) (d) \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})\)

Let \(\mathbf{a}=\langle 3,-1\rangle, \mathbf{b}=\langle 1,-1\rangle\), and \(\mathbf{c}=\langle 0,5\rangle .\) Find each of the following: (a) \(-4 \mathbf{a}+3 \mathbf{b}\) (b) \(\mathbf{b} \cdot \mathbf{c}\) (c) \((\mathbf{a}+\mathbf{b}) \cdot \mathbf{c}\) (d) \(2 \mathbf{c} \cdot(3 \mathbf{a}+4 \mathbf{b})\) (e) \(\|\mathbf{b}\| \mathbf{b} \cdot \mathbf{a}\) (f) \(\|\mathbf{c}\|^{2}-\mathbf{c} \cdot \mathbf{c}\)

Name and sketch the graph of each of the following equations in three-space. $$ 3 x+2 z=10 $$

Change the following from cylindrical to Cartesian (rectangular) coordinates. (a) \((6, \pi / 6,-2)\) (b) \((4,4 \pi / 3,-8)\)

sketch the curve over the indicated domain for \(t\). Find \(\mathbf{v}, \mathbf{a}, \mathbf{T}\), and \(\kappa\) at the point where \(t=t_{1} .\) $$ \mathbf{r}(t)=t \mathbf{i}+2 \cos t \mathbf{j}+2 \sin t \mathbf{k} ; \quad 0 \leq t \leq 4 \pi ; t_{1}=\pi $$

Find the parametric equations of the line through the given pair of points. \((4,2,3),(6,2,-1)\)

Find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow 1}\left[\frac{t-1}{t^{2}-1} \mathbf{i}-\frac{t^{2}+2 t-3}{t-1} \mathbf{j}\right] $$

Find all vectors perpendicular to both of the vectors \(\mathbf{a}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}\) and \(\mathbf{b}=-2 \mathbf{i}+2 \mathbf{j}-4 \mathbf{k}\)

Find the cosine of the angle between \(\mathbf{a}\) and \(\mathbf{b}\) and make a sketch. (a) \(\mathbf{a}=\langle 1,-3\rangle, \mathbf{b}=\langle-1,2\rangle\) (b) \(\mathbf{a}=\langle-1,-2\rangle, \mathbf{b}=\langle 6,0\rangle\) (c) \(\mathbf{a}=\langle 2,-1\rangle, \mathbf{b}=\langle-2,-4\rangle\) (d) \(\mathbf{a}=\langle 4,-7\rangle, \mathbf{b}=\langle-8,10\rangle\)

What is peculiar to the coordinates of all points in the \(y z\) -plane? On the \(z\) -axis?

Name and sketch the graph of each of the following equations in three-space. $$ z^{2}=3 y $$

Change the following from spherical to Cartesian coordinates. (a) \((8, \pi / 4, \pi / 6)\) (b) \((4, \pi / 3,3 \pi / 4)\)

sketch the curve over the indicated domain for \(t\). Find \(\mathbf{v}, \mathbf{a}, \mathbf{T}\), and \(\kappa\) at the point where \(t=t_{1} .\) $$ \mathbf{r}(t)=5 \cos t \mathbf{i}+2 t \mathbf{j}+5 \sin t \mathbf{k} ; \quad 0 \leq t \leq 4 \pi ; t_{1}=\pi $$

Find the parametric equations of the line through the given pair of points. \((5,-3,-3),(5,4,2)\)

Find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow-2}\left[\frac{2 t^{2}-10 t-28}{t+2} \mathbf{i}-\frac{7 t^{3}}{t-3} \mathbf{j}\right] $$

Find all vectors perpendicular to both of the vectors \(\mathbf{a}=-2 \mathbf{i}+5 \mathbf{j}-2 \mathbf{k}\) and \(\mathbf{b}=3 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}\)

Find the angle between \(\mathbf{a}\) and \(\mathbf{b}\) and make a sketch. (a) \(\mathbf{a}=12 \mathbf{i}, \mathbf{b}=-5 \mathbf{i}\) (b) \(\mathbf{a}=4 \mathbf{i}+3 \mathbf{j}, \mathbf{b}=-8 \mathbf{i}-6 \mathbf{j}\) (c) \(\mathbf{a}=-\mathbf{i}+3 \mathbf{j}, \mathbf{b}=2 \mathbf{i}-6 \mathbf{j}\) (d) \(\mathbf{a}=\sqrt{3} \mathbf{i}+\mathbf{j}, \mathbf{b}=3 \mathbf{i}+\sqrt{3} \mathbf{j}\)

What is peculiar to the coordinates of all points in the \(x z\) -plane? On the \(y\) -axis?

Change the following from Cartesian to spherical coordinates. (a) \((2,-2 \sqrt{3}, 4)\) (b) \((-\sqrt{2}, \sqrt{2}, 2 \sqrt{3})\)

sketch the curve over the indicated domain for \(t\). Find \(\mathbf{v}, \mathbf{a}, \mathbf{T}\), and \(\kappa\) at the point where \(t=t_{1} .\) $$ \mathbf{r}(t)=\frac{t^{2}}{8} \mathbf{i}+5 \cos t \mathbf{j}+5 \sin t \mathbf{k} ; \quad 0 \leq t \leq 4 \pi ; t_{1}=\pi $$

Write both the parametric equations and the symmetric equations for the line through the given point parallel to the given vector. \((4,5,6),\langle 3,2,1\rangle\)

Find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow 0}\left[\frac{\sin t \cos t}{t} \mathbf{i}-\frac{7 t^{3}}{e^{t}} \mathbf{j}+\frac{t}{t+1} \mathbf{k}\right] $$

Find the unit vectors perpendicular to the plane determined by the three points \((1,3,5),(3,-1,2)\), and \((4,0,1)\).

Let \(\mathbf{a}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}, \mathbf{b}=\mathbf{j}+\mathbf{k}\), and \(\mathbf{c}=-\mathbf{i}+\mathbf{j}+2 \mathbf{k} .\) Find each of the following: (a) \(\mathbf{a} \cdot \mathbf{b}\) (b) \((\mathbf{a}+\mathbf{c}) \cdot \mathbf{b}\) (c) \(\mathbf{a} /\|\mathbf{a}\|\) (d) \((\mathbf{b}-\mathbf{c}) \cdot \mathbf{a}\) (e) \(\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}\) (f) \(\mathbf{b} \cdot \mathbf{b}-\|\mathbf{b}\|^{2}\)

Find the distance between the following pairs of points. (a) \((6,-1,0)\) and \((1,2,3)\) (b) \((-2,-2,0)\) and \((2,-2,-3)\) (c) \((e, \pi, 0)\) and \((-\pi,-4, \sqrt{3})\)

Name and sketch the graph of each of the following equations in three-space. $$ 2 x^{2}-16 z^{2}=0 $$

Change the following from Cartesian to cylindrical coordinates. (a) \((2,2,3)\) (b) \((4 \sqrt{3},-4,6)\)

sketch the curve over the indicated domain for \(t\). Find \(\mathbf{v}, \mathbf{a}, \mathbf{T}\), and \(\kappa\) at the point where \(t=t_{1} .\) $$ \mathbf{r}(t)=\frac{t^{2}}{4} \mathbf{i}+2 \cos t \mathbf{j}+2 \sin t \mathbf{k} ; \quad 0 \leq t \leq 4 \pi ; t_{1}=\pi $$

Write both the parametric equations and the symmetric equations for the line through the given point parallel to the given vector. \((-1,3,-6),\langle-2,0,5\rangle\)

Find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow \infty}\left[\frac{t \sin t}{t^{2}} \mathbf{i}-\frac{7 t^{3}}{t^{3}-3 t} \mathbf{j}-\frac{\sin t}{t} \mathbf{k}\right] $$

Find the unit vectors perpendicular to the plane determined by the three points \((-1,3,0),(5,1,2)\), and \((4,-3,-1)\).

Let \(\mathbf{a}=\langle\sqrt{2}, \sqrt{2}, 0\rangle, \mathbf{b}=\langle 1,-1,1\rangle\), and \(\mathbf{c}=\langle-2,2,1\rangle\). Find each of the following: (a) \(\mathbf{a} \cdot \mathbf{c}\) (b) \((\mathbf{a}-\mathbf{c}) \cdot \mathbf{b}\) (c) \(\mathbf{a} /\|\mathbf{a}\|\) (d) \((\mathbf{b}-\mathbf{c}) \cdot \mathbf{a}\) (e) \(\frac{\mathbf{b} \cdot \mathbf{c}}{\|\mathbf{b}\|\|\mathbf{c}\|}\) (f) \(\mathbf{a} \cdot \mathbf{a}-\|\mathbf{a}\|^{2}\)

Show that \((4,5,3),(1,7,4)\), and \((2,4,6)\) are vertices of an equilateral triangle.

Name and sketch the graph of each of the following equations in three-space. $$ 4 x^{2}+9 y^{2}+49 z^{2}=1764 $$

Sketch the graph of the given cylindrical or spherical equation. $$ r=5 $$

Write both the parametric equations and the symmetric equations for the line through the given point parallel to the given vector. \((1,1,1),\langle-10,-100,-1000\rangle\)

Find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow 0^{+}}\left\langle\ln \left(t^{3}\right), t^{2} \ln t, t\right\rangle $$

Find the area of the parallelogram with \(\mathbf{a}=-\mathbf{i}+\mathbf{j}-3 \mathbf{k}\) and \(\mathbf{b}=4 \mathbf{i}+2 \mathbf{j}-4 \mathbf{k}\) as the adjacent sides.

Show that \((2,1,6),(4,7,9)\), and \((8,5,-6)\) are vertices of a right triangle. Hint: Only right triangles satisfy the Pythagorean Theorem.