<< Chapter < Page | Chapter >> Page > |
Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point $A(2,3,5)$ is the opposite vertex to the origin, then find
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)
$\left(z-2\right)\left(z-5\right)=0$
${\left(y-1\right)}^{2}+{(z-1)}^{2}=1$
A cylinder of radius
$1$ centered on the line
$y=1,z=1$
${\left(x-2\right)}^{2}+{(z-5)}^{2}=4$
Write the equation of the plane passing through point $(1,1,1)$ that is parallel to the xy -plane.
$z=1$
Write the equation of the plane passing through point $(1,\mathrm{-3},2)$ that is parallel to the xz -plane.
Find an equation of the plane passing through points $(1,\mathrm{-3},\mathrm{-2}),$ $(0,3,\mathrm{-2}),$ and $(1,0,\mathrm{-2}).$
$z=\mathrm{-2}$
Find an equation of the plane passing through points $(1,9,2),$ $(1,3,6),$ and $(1,\mathrm{-7},8).$
For the following exercises, find the equation of the sphere in standard form that satisfies the given conditions.
Center $C\left(\mathrm{-1},7,4\right)$ and radius $4$
${(x+1)}^{2}+{(y-7)}^{2}+{(z-4)}^{2}=16$
Center $C\left(\mathrm{-4},7,2\right)$ and radius $6$
Diameter $PQ,$ where $P\left(\mathrm{-1},5,7\right)$ and $Q\left(\mathrm{-5},2,9\right)$
${(x+3)}^{2}+{(y-3.5)}^{2}+{(z-8)}^{2}=\frac{29}{4}$
Diameter $PQ,$ where $P\left(\mathrm{-16},\mathrm{-3},9\right)$ and $Q\left(\mathrm{-2},3,5\right)$
For the following exercises, find the center and radius of the sphere with an equation in general form that is given.
$P(1,2,3)$ ${x}^{2}+{y}^{2}+{z}^{2}-4z+3=0$
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 $\overrightarrow{PQ}$ with the initial point at $P$ and the terminal point at $Q$
$P\left(3,0,2\right)$ and $Q\left(\mathrm{-1},\mathrm{-1},4\right)$
a. $\overrightarrow{PQ}=\u27e8\mathrm{-4},\mathrm{-1},2\u27e9;$ b. $\overrightarrow{PQ}=\mathrm{-4}\text{i}-\text{j}+2\text{k}$
$P\left(0,10,5\right)$ and $Q\left(1,1,\mathrm{-3}\right)$
$P\left(\mathrm{-2},5,\mathrm{-8}\right)$ and $M\left(1,\mathrm{-7},4\right),$ where $M$ is the midpoint of the line segment $PQ$
a. $\overrightarrow{PQ}=\u27e86,\mathrm{-24},24\u27e9;$ b. $\overrightarrow{PQ}=6\text{i}-24\mathbf{\text{j}}+24\text{k}$
$Q\left(0,7,\mathrm{-6}\right)$ and $M\left(\mathrm{-1},3,2\right),$ where $M$ is the midpoint of the line segment $PQ$
Find terminal point $Q$ of vector $\overrightarrow{PQ}=\u27e87,\mathrm{-1},3\u27e9$ with the initial point at $P\left(\mathrm{-2},3,5\right).$
$Q(5,2,8)$
Find initial point $P$ of vector $\overrightarrow{PQ}=\u27e8\mathrm{-9},1,2\u27e9$ with the terminal point at $Q\left(10,0,\mathrm{-1}\right).$
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 $\mathrm{-5}\text{a}+3\text{b}$ in component form.
$\text{a}=\u27e8\mathrm{-1},\mathrm{-2},4\u27e9,$ $\text{b}=\u27e8\mathrm{-5},6,\mathrm{-7}\u27e9$
$\text{a}+\mathbf{\text{b}}=\u27e8\mathrm{-6},4,\mathrm{-3}\u27e9,$ $4\text{a}=\u27e8\mathrm{-4},\mathrm{-8},16\u27e9,$ $\mathrm{-5}\text{a}+3\text{b}=\u27e8\mathrm{-10},28,\mathrm{-41}\u27e9$
$\text{a}=\u27e83,\mathrm{-2},4\u27e9,$ $\text{b}=\u27e8\mathrm{-5},6,\mathrm{-9}\u27e9$
$\text{a}=\text{\u2212}\text{k},$ $\text{b}=\text{\u2212}\mathbf{\text{i}}$
$\text{a}+\mathbf{\text{b}}=\u27e8\mathrm{-1},0,\mathrm{-1}\u27e9,$ $4\text{a}=\u27e80,0,\mathrm{-4}\u27e9,$ $\mathrm{-5}\text{a}+3\text{b}=\u27e8\mathrm{-3},0,5\u27e9$
$\text{a}=\text{i}+\text{j}+\text{k},$ $\text{b}=2\text{i}-3\mathbf{\text{j}}+2\text{k}$
For the following exercises, vectors u and v are given. Find the magnitudes of vectors $\text{u}-\text{v}$ and $\mathrm{-2}\text{u}.$
$\text{u}=2\text{i}+3\text{j}+4\text{k},$ $\text{v}=\text{\u2212}\text{i}+5\text{j}-\text{k}$
$\Vert \text{u}-\text{v}\Vert =\sqrt{38},$ $\Vert \mathrm{-2}\text{u}\Vert =2\sqrt{29}$
$\text{u}=\text{i}+\text{j},$ $\text{v}=\text{j}-\text{k}$
$\text{u}=\u27e82\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t,\mathrm{-2}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t,3\u27e9,$ $\text{v}=\u27e80,0,3\u27e9,$ where $t$ is a real number.
$\Vert \text{u}-\text{v}\Vert =2,$ $\Vert \mathrm{-2}\text{u}\Vert =2\sqrt{13}$
$\text{u}=\u27e80,1,\phantom{\rule{0.2em}{0ex}}\text{sinh}\phantom{\rule{0.2em}{0ex}}t\u27e9,$ $\text{v}=\u27e81,1,0\u27e9,$ 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.
$\text{a}=3\text{i}-4\text{j}$
$\text{a}=\frac{3}{5}\text{i}-\frac{4}{5}\text{j}$
$\text{a}=\u27e84,\mathrm{-3},6\u27e9$
$\text{a}=\overrightarrow{PQ},$ where $P\left(\mathrm{-2},3,1\right)$ and $Q\left(0,\mathrm{-4},4\right)$
$\u27e8\frac{2}{\sqrt{62}},-\frac{7}{\sqrt{62}},\frac{3}{\sqrt{62}}\u27e9$
$\text{a}=\overrightarrow{OP},$ where $P\left(\mathrm{-1},\mathrm{-1},1\right)$
$\text{a}=\text{u}-\text{v}+\mathbf{\text{w}},$ where $\text{u}=\text{i}-\text{j}-\text{k},$ $\text{v}=2\text{i}-\text{j}+\text{k},$ and $\text{w}=\text{\u2212}\text{i}+\text{j}+3\text{k}$
$\u27e8-\frac{2}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}\u27e9$
Notification Switch
Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?