# 2.2 Vectors in three dimensions  (Page 7/14)

 Page 7 / 14

## Key equations

• Distance between two points in space:
$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}+{\left({z}_{2}-{z}_{1}\right)}^{2}}$
• Sphere with center $\left(a,b,c\right)$ and radius r :
${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}+{\left(z-c\right)}^{2}={r}^{2}$

Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point $A\left(2,3,5\right)$ is the opposite vertex to the origin, then find

1. the coordinates of the other six vertices of the box and
2. the length of the diagonal of the box determined by the vertices $O$ and $A.$

a. $\left(2,0,5\right),\left(2,0,0\right),\left(2,3,0\right),\left(0,3,0\right),\left(0,3,5\right),\left(0,0,5\right);$ b. $\sqrt{38}$

Find the coordinates of point $P$ and determine its distance to the origin.

For the following exercises, describe and graph the set of points that satisfies the given equation.

$\left(y-5\right)\left(z-6\right)=0$

A union of two planes: $y=5$ (a plane parallel to the xz -plane) and $z=6$ (a plane parallel to the xy -plane)

$\left(z-2\right)\left(z-5\right)=0$

${\left(y-1\right)}^{2}+{\left(z-1\right)}^{2}=1$

A cylinder of radius $1$ centered on the line $y=1,z=1$

${\left(x-2\right)}^{2}+{\left(z-5\right)}^{2}=4$

Write the equation of the plane passing through point $\left(1,1,1\right)$ that is parallel to the xy -plane.

$z=1$

Write the equation of the plane passing through point $\left(1,-3,2\right)$ that is parallel to the xz -plane.

Find an equation of the plane passing through points $\left(1,-3,-2\right),$ $\left(0,3,-2\right),$ and $\left(1,0,-2\right).$

$z=-2$

Find an equation of the plane passing through points $\left(1,9,2\right),$ $\left(1,3,6\right),$ and $\left(1,-7,8\right).$

For the following exercises, find the equation of the sphere in standard form that satisfies the given conditions.

Center $C\left(-1,7,4\right)$ and radius $4$

${\left(x+1\right)}^{2}+{\left(y-7\right)}^{2}+{\left(z-4\right)}^{2}=16$

Center $C\left(-4,7,2\right)$ and radius $6$

Diameter $PQ,$ where $P\left(-1,5,7\right)$ and $Q\left(-5,2,9\right)$

${\left(x+3\right)}^{2}+{\left(y-3.5\right)}^{2}+{\left(z-8\right)}^{2}=\frac{29}{4}$

Diameter $PQ,$ where $P\left(-16,-3,9\right)$ and $Q\left(-2,3,5\right)$

For the following exercises, find the center and radius of the sphere with an equation in general form that is given.

$P\left(1,2,3\right)$ ${x}^{2}+{y}^{2}+{z}^{2}-4z+3=0$

Center $C\left(0,0,2\right)$ and radius $1$

${x}^{2}+{y}^{2}+{z}^{2}-6x+8y-10z+25=0$

For the following exercises, express vector $\stackrel{\to }{PQ}$ with the initial point at $P$ and the terminal point at $Q$

1. in component form and
2. by using standard unit vectors.

$P\left(3,0,2\right)$ and $Q\left(-1,-1,4\right)$

a. $\stackrel{\to }{PQ}=⟨-4,-1,2⟩;$ b. $\stackrel{\to }{PQ}=-4\text{i}-\text{j}+2\text{k}$

$P\left(0,10,5\right)$ and $Q\left(1,1,-3\right)$

$P\left(-2,5,-8\right)$ and $M\left(1,-7,4\right),$ where $M$ is the midpoint of the line segment $PQ$

a. $\stackrel{\to }{PQ}=⟨6,-24,24⟩;$ b. $\stackrel{\to }{PQ}=6\text{i}-24\mathbf{\text{j}}+24\text{k}$

$Q\left(0,7,-6\right)$ and $M\left(-1,3,2\right),$ where $M$ is the midpoint of the line segment $PQ$

Find terminal point $Q$ of vector $\stackrel{\to }{PQ}=⟨7,-1,3⟩$ with the initial point at $P\left(-2,3,5\right).$

$Q\left(5,2,8\right)$

Find initial point $P$ of vector $\stackrel{\to }{PQ}=⟨-9,1,2⟩$ with the terminal point at $Q\left(10,0,-1\right).$

For the following exercises, use the given vectors $\text{a}$ and $\text{b}$ to find and express the vectors $\text{a}+\mathbf{\text{b}},$ $4\text{a},$ and $-5\text{a}+3\text{b}$ in component form.

$\text{a}=⟨-1,-2,4⟩,$ $\text{b}=⟨-5,6,-7⟩$

$\text{a}+\mathbf{\text{b}}=⟨-6,4,-3⟩,$ $4\text{a}=⟨-4,-8,16⟩,$ $-5\text{a}+3\text{b}=⟨-10,28,-41⟩$

$\text{a}=⟨3,-2,4⟩,$ $\text{b}=⟨-5,6,-9⟩$

$\text{a}=\text{−}\text{k},$ $\text{b}=\text{−}\mathbf{\text{i}}$

$\text{a}+\mathbf{\text{b}}=⟨-1,0,-1⟩,$ $4\text{a}=⟨0,0,-4⟩,$ $-5\text{a}+3\text{b}=⟨-3,0,5⟩$

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

For the following exercises, vectors u and v are given. Find the magnitudes of vectors $\text{u}-\text{v}$ and $-2\text{u}.$

$\text{u}=2\text{i}+3\text{j}+4\text{k},$ $\text{v}=\text{−}\text{i}+5\text{j}-\text{k}$

$‖\text{u}-\text{v}‖=\sqrt{38},$ $‖-2\text{u}‖=2\sqrt{29}$

$\text{u}=\text{i}+\text{j},$ $\text{v}=\text{j}-\text{k}$

$\text{u}=⟨2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t,-2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t,3⟩,$ $\text{v}=⟨0,0,3⟩,$ where $t$ is a real number.

$‖\text{u}-\text{v}‖=2,$ $‖-2\text{u}‖=2\sqrt{13}$

$\text{u}=⟨0,1,\phantom{\rule{0.2em}{0ex}}\text{sinh}\phantom{\rule{0.2em}{0ex}}t⟩,$ $\text{v}=⟨1,1,0⟩,$ where $t$ is a real number.

For the following exercises, find the unit vector in the direction of the given vector $\text{a}$ and express it using standard unit vectors.

$\text{a}=3\text{i}-4\text{j}$

$\text{a}=\frac{3}{5}\text{i}-\frac{4}{5}\text{j}$

$\text{a}=⟨4,-3,6⟩$

$\text{a}=\stackrel{\to }{PQ},$ where $P\left(-2,3,1\right)$ and $Q\left(0,-4,4\right)$

$⟨\frac{2}{\sqrt{62}},-\frac{7}{\sqrt{62}},\frac{3}{\sqrt{62}}⟩$

$\text{a}=\stackrel{\to }{OP},$ where $P\left(-1,-1,1\right)$

$\text{a}=\text{u}-\text{v}+\mathbf{\text{w}},$ where $\text{u}=\text{i}-\text{j}-\text{k},$ $\text{v}=2\text{i}-\text{j}+\text{k},$ and $\text{w}=\text{−}\text{i}+\text{j}+3\text{k}$

$⟨-\frac{2}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}⟩$

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