벡터와 공간 전체 문제

[T] The “bumpy sphere” with an equation in spherical coordinates is $\rho =a+b\,\cos (m\theta )\sin (n\varphi ),$ with $\theta \in [0,2\pi ]$ and $\varphi \in [0,\pi ],$ where $a$ and $b$ are positive numbers and $m$ and $n$ are positive integers, may be used in applied mathematics to model tumor growth.

a. Show that the “bumpy sphere” is contained inside a sphere of equation $\rho =a+b.$ Find the values of $\theta$ and $\varphi$ at which the two surfaces intersect. b. Use a CAS to graph the surface for $a=14,$ $b=2,$ $m=4,$ and $n=6$ along with sphere $\rho =a+b.$ c. Find an equation of the intersection curve of the surface at b. with the cone $\varphi =\frac{\pi }{12}.$ Graph the intersection curve in the plane of intersection.

Chapter Review Exercises

For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample.

For vectors $\text{a}$ and $\text{b}$ and any given scalar $c,$ $c(\text{a}\cdot \text{b})=(c\text{a})\cdot \text{b}.$

Chapter Review Exercises

For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample.

For vectors $\text{a}$ and $\text{b}$ and any given scalar $c,$ $c(\text{a}\,\times \,\text{b})=(c\text{a})\,\times \,\text{b}.$

The symmetric equation for the line of intersection between two planes $x+y+z=2$ and $x+2y-4z=5$ is given by $-\frac{x-1}{6}=\frac{y-1}{5}=z.$

Chapter Review Exercises

For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample.

If $\text{a}\cdot \text{b}=0,$ then $\text{a}$ is perpendicular to $\text{b}.$

For the following exercises, use the given vectors to find the quantities.

$\text{a}=9\text{i}-2\text{j},\text{b}=-3\text{i}+\text{j}$

a. $3\text{a}+\text{b}$ b. $\| \text{a}\|$ c. $\text{a}\,\times \,\| \text{b}\,\times \,\text{c}\|$ d. $\text{b}\,\cdot \,\text{a}$

For the following exercises, use the given vectors to find the quantities.

$\text{a}=2\text{i}+\text{j}-9\text{k},\text{b}=-\text{i}+2\text{k},\text{c}=4\text{i}-2\text{j}+\text{k}$

a. $2\text{a}-\text{b}$ b. $\text{b}\,\times \,\text{c}$ c. $\text{b}\,\times \,\| \text{b}\,\times \,\text{c}\|$ d. $\text{c}\,\times \,\| \text{b}\,\times \,\text{a}\|$ e. ${\text{proj}}_{\text{a}}\text{b}$

Find the values of $a$ such that vectors $\langle 2,4,a\rangle$ and $\langle 0,-1,a\rangle$ are orthogonal.

Find the unit vector that has the same direction as vector $\text{v}$ that begins at $(0,-3)$ and ends at $(4,10).$

Find the unit vector that has the same direction as vector $\text{v}$ that begins at $(1,4,10)$ and ends at $(3,0,4).$

For the following exercises, find the area or volume of the given shapes.

The parallelogram spanned by vectors $\text{a}=\langle 1,13\rangle \,\text{and}\,\text{b}=\langle 3,21\rangle$

For the following exercises, find the area or volume of the given shapes.

The parallelepiped formed by $\text{a}=\langle 1,4,1\rangle \,\text{and}\,\text{b}=\langle 3,6,2\rangle ,$ and $\text{c}=\langle -2,1,-5\rangle$

The line that passes through point $(2,-3,7)$ that is parallel to vector $\langle 1,3,-2\rangle$

For the following exercises, find the vector and parametric equations of the line with the given properties.

The line that passes through points $(1,3,5)$ and $(-2,6,-3)$

The plane that passes through point $(4,7,-1)$ and has normal vector $\text{n}=\langle 3,4,2\rangle$

For the following exercises, find an equation of the plane with the given properties.

The plane that passes through points $(0,1,5),(2,-1,6),\,\text{and}\,(3,2,5).$

For the following exercises, find the traces for the surfaces in planes $x=k,y=k,\,\text{and}\,z=k.$ Then, describe and draw the surfaces.

$9{x}^{2}+4{y}^{2}-16y+36{z}^{2}=20$

For the following exercises, find the traces for the surfaces in planes $x=k,y=k,\,\text{and}\,z=k.$ Then, describe and draw the surfaces.

${x}^{2}={y}^{2}+{z}^{2}$

For the following exercises, write the given equation in cylindrical coordinates and spherical coordinates.

${x}^{2}+{y}^{2}+{z}^{2}=144$

For the following exercises, write the given equation in cylindrical coordinates and spherical coordinates.

$z={x}^{2}+{y}^{2}-1$