벡터와 공간 전체 문제

Nonzero vectors $\text{u}$ and $\text{v}$ are called collinear if there exists a nonzero scalar $\alpha$ such that $\text{v}=\alpha \text{u}.$ Show that $\text{u}$ and $\text{v}$ are collinear if and only if $\text{u}\,\times \,\text{v}=0.$

Nonzero vectors $\text{u}$ and $\text{v}$ are called collinear if there exists a nonzero scalar $\alpha$ such that $\text{v}=\alpha \text{u}.$ Show that vectors $\vec{AB}$ and $\vec{AC}$ are collinear, where $A(4,1,0),$ $B(6,5,-2),$ and $C(5,3,-1).$

Let $\text{u},\text{v},\,\text{and}\,\text{w}$ be three-dimensional vectors and $c$ be a real number. Prove the following properties of the cross product.

a. $\text{u}\,\times \,\text{u}=0$ b. $\text{u}\,\times \,(\text{v}+\text{w})=(\text{u}\,\times \,\text{v})+(\text{u}\,\times \,\text{w})$ c. $c(\text{u}\,\times \,\text{v})=(c\text{u})\,\times \,\text{v}=\text{u}\,\times \,(c\text{v})$ d. $\text{u}\cdot (\text{u}\,\times \,\text{v})=0$

Show that vectors $\text{u}=\langle 1,0,-8\rangle ,$ $\text{v}=\langle 0,1,6\rangle ,$ and $\text{w}=\langle -1,9,3\rangle$ satisfy the following properties of the cross product.

a. $\text{u}\,\times \,\text{u}=0$ b. $\text{u}\,\times \,(\text{v}+\text{w})=(\text{u}\,\times \,\text{v})+(\text{u}\,\times \,\text{w})$ c. $c(\text{u}\,\times \,\text{v})=(c\text{u})\,\times \,\text{v}=\text{u}\,\times \,(c\text{v})$ d. $\text{u}\cdot (\text{u}\,\times \,\text{v})=0$

Nonzero vectors $\text{u},\text{v},\,\text{and}\,\text{w}$ are said to be linearly dependent if one of the vectors is a linear combination of the other two. For instance, there exist two nonzero real numbers $\alpha$ and $\beta$ such that $\text{w}=\alpha \text{u}+\beta \text{v}.$ Otherwise, the vectors are called linearly independent. Show that $\text{u},\text{v},\,\text{and}\,\text{w}$ are coplanar if and only if they are linear dependent.

Consider vectors $\text{u}=\langle 1,4,-7\rangle ,$ $\text{v}=\langle 2,-1,4\rangle ,$ $\text{w}=\langle 0,-9,18\rangle ,$ and $\text{p}=\langle 0,-9,17\rangle .$

a. Show that $\text{u},\text{v},\,\text{and}\,\text{w}$ are coplanar by using their triple scalar product b. Show that $\text{u},\text{v},\,\text{and}\,\text{w}$ are coplanar, using the definition that there exist two nonzero real numbers $\alpha$ and $\beta$ such that $\text{w}=\alpha \text{u}+\beta \text{v}.$ c. Show that $\text{u},\text{v},\,\text{and}\,\text{p}$ are linearly independent—that is, none of the vectors is a linear combination of the other two.

Show that vectors $\text{i}+\text{j},$ $\text{i}-\text{j},$ and $\text{i}+\text{j}+\text{k}$ are linearly independent—that is, there do not exist two nonzero real numbers $\alpha$ and $\beta$ such that $\text{i}+\text{j}+\text{k}=\alpha (\text{i}+\text{j})+\beta (\text{i}-\text{j}).$

Consider $\text{u}$ and $\text{v}$ two three-dimensional vectors. If the magnitude of the cross product vector $\text{u}\,\times \,\text{v}$ is $k$ times larger than the magnitude of vector $\text{u},$ show that the magnitude of $\text{v}$ is greater than or equal to $k,$ where $k$ is a natural number.

Show that the line passing through points $P(3,1,0)$ and $Q(1,4,-3)$ is perpendicular to the line with equations $x = 3 + 3t, y = 1 + 8t, z = 6t,$ $t\in \mathbb{R}.$

For the following exercises, the equation of a plane is given.

a. Find normal vector $\text{n}$ to the plane. Express $\text{n}$ using standard unit vectors. b. Find the intersections of the plane with the coordinate axes. c. Sketch the plane.

Show that line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ is parallel to plane $x-2y+z=6.$

Consider the planes of equations $x+y+z=1$ and $x+z=0.$

a. Show that the planes intersect. b. Find symmetric equations of the line passing through point $P(1,4,6)$ that is parallel to the line of intersection of the planes.

Consider the planes of equations $-y+z-2=0$ and $x-y=0.$

a. Show that the planes intersect. b. Find parametric equations of the line passing through point $P(-8,0,2)$ that is parallel to the line of intersection of the planes.

Show that the lines of equations $x=t,y=1+t,z=2+t,$ $t\in \mathbb{R},$ and $\frac{x}{2}=\frac{y-1}{3}=z-3$ are skew, and find the distance between them.

Show that the lines of equations $x=-1+t,y=-2+t,z=3t,$ $t\in \mathbb{R},$ and $x=5+s,y=-8+2s,z=7s,$ $s\in \mathbb{R}$ are skew, and find the distance between them.

[T] A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) $A(8,0,0),$ $B(8,18,0),$ $C(0,18,8),$ and $D(0,0,8)$ (see the following figure).

a. Find the general form of the equation of the plane that contains the solar panel by using points $A,B,\,\text{and}\,C,$ and show that its normal vector is equivalent to $\vec{AB}\,\times \,\vec{AD}.$ b. Find parametric equations of line ${L}_{1}$ that passes through the center of the solar panel and has direction vector $\text{s}=\frac{1}{\sqrt{3}}\text{i}+\frac{1}{\sqrt{3}}\text{j}+\frac{1}{\sqrt{3}}\text{k},$ which points toward the position of the Sun at a particular time of day. c. Find symmetric equations of line ${L}_{2}$ that passes through the center of the solar panel and is perpendicular to it. d. Determine the angle of elevation of the Sun above the solar panel by using the angle between lines ${L}_{1}$ and ${L}_{2}.$

Show that quadric surface ${x}^{2}+{y}^{2}+{z}^{2}+2xy+2xz+2yz+x+y+z=0$ reduces to two parallel planes.

Show that quadric surface ${x}^{2}+{y}^{2}+{z}^{2}-2xy-2xz+2yz-1=0$ reduces to two parallel planes passing.

[T] Consider the torus of equation ${({x}^{2}+{y}^{2}+{z}^{2}+{R}^{2}-{r}^{2})}^{2}=4{R}^{2}({x}^{2}+{y}^{2}),$ where $R\ge r>0.$

a. Write the equation of the torus in spherical coordinates. b. If $R=r,$ the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is $\rho =2R\,\sin \,\varphi .$ c. Use a CAS to graph the horn torus with $R=r=2$ in spherical coordinates.

[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.

Consider the curve described by the vector-valued function $\text{r}(t)=(50{e}^{-t}\cos t)\text{i}+(50{e}^{-t}\sin t)\text{j}+(5-5{e}^{-t})\text{k}.$

Eliminate the parameter t to show that $z=5-\frac{r}{10}$ where ${r}^{2}={x}^{2}+{y}^{2}.$