Chapter 11

Let \(\mathbf{a}=-3 \mathbf{i}+2 \mathbf{j}-2 \mathbf{k}, \quad \mathbf{b}=-\mathbf{i}+2 \mathbf{j}-4 \mathbf{k}\), 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) \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})\)

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

$$ \lim _{t \rightarrow 1}\left[2 t \mathbf{i}-t^{2} \mathbf{j}\right] $$

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

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) \(\mathbf{b} \cdot \mathbf{b}-\|\mathbf{b}\|\)

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

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) \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})\)

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 \(t=t_{1}\). \(\mathbf{r}(t)=t^{2} \mathbf{i}+(2 t+1) \mathbf{j} ; \quad 0 \leq t \leq 2 ; t_{1}=1\)

$$ \lim _{t \rightarrow 3}\left[2(t-3)^{2} \mathbf{i}-7 t^{3} \mathbf{j}\right] $$

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

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) \(-4 a+3 b\) (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) \(\|\mathbf{c}\|^{2}-\mathbf{c} \cdot \mathbf{c}\)

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

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

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 \(t=t_{1}\). \(\mathbf{r}(t)=t \mathbf{i}+2 \cos t \mathbf{j}+2 \sin t \mathbf{k} ; \quad 0 \leq t \leq 4 \pi ; t_{1}=\pi\)

$$ \lim _{t \rightarrow 1}\left[\frac{t-1}{t^{2}-1} \mathbf{i}-\frac{t^{2}+2 t-3}{t-1} \mathbf{j}\right] $$

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

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?

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

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

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 \(t=t_{1}\). \(\mathbf{r}(t)=5 \cos t \mathbf{i}+2 t \mathbf{j}+5 \sin t \mathbf{k} ; \quad 0 \leq t \leq 4 \pi ; t_{1}=\pi\)

$$ \lim _{t \rightarrow-2}\left[\frac{2 t^{2}-10 t-28}{t+2} \mathbf{i}-\frac{7 t^{3}}{t-3} \mathbf{j}\right] $$

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

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?

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

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

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 \(t=t_{1}\). \(\mathbf{r}(t)=\frac{t^{2}}{8} \mathbf{i}+5 \cos t \mathbf{j}+5 \sin t \mathbf{k} ; \quad 0 \leq t \leq 4 \pi ; t_{1}=\pi\)

$$ \lim _{t \rightarrow 0}\left[\frac{\sin t \cos t}{t} \mathbf{i}-\frac{7 t^{3}}{e^{t}} \mathbf{j}+\frac{t}{t+1} \mathbf{k}\right] $$

Name and sketch the graph of each of the following equations in three-space. $$ x^{2}+y^{2}-8 x+4 y+13=0 $$

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 \((-\pi,-4, \sqrt{3})\)

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

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 \(t=t_{1}\). \(\mathbf{r}(t)=\frac{t^{2}}{4} \mathbf{i}+2 \cos t \mathbf{j}+2 \sin t \mathbf{k} ; \quad 0 \leq t \leq 4 \pi ; t_{1}=\pi\)

$$ \lim _{t \rightarrow \infty}\left[\frac{t \sin t}{t^{2}} \mathbf{i}-\frac{7 t^{3}}{t^{3}-3 t} \mathbf{j}-\frac{\sin t}{t} \mathbf{k}\right] $$

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

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) \(a^{\cdot} \mathbf{c}\) (b) \((a-c) \cdot b\) (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.

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

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.

$$ \lim _{t \rightarrow 0^{+}}\left(\ln \left(t^{3}\right), t^{2} \ln t, t\right) $$

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

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.

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