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For the following exercises, evaluate each root.
Evaluate the cube root of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=64\mathrm{cis}\left(\mathrm{210\xb0}\right).$
Evaluate the square root of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=25\mathrm{cis}\left(\frac{3\pi}{2}\right).$
$5\mathrm{cis}\left(\frac{3\pi}{4}\right),5\mathrm{cis}\left(\frac{7\pi}{4}\right)$
For the following exercises, plot the complex number in the complex plane.
For the following exercises, eliminate the parameter $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ to rewrite the parametric equation as a Cartesian equation.
$\{\begin{array}{l}x\left(t\right)=3t-1\hfill \\ y\left(t\right)=\sqrt{t}\hfill \end{array}$
$\{\begin{array}{l}x(t)=-\mathrm{cos}\text{\hspace{0.17em}}t\hfill \\ y(t)=2{\mathrm{sin}}^{2}t\hfill \end{array}$
${x}^{2}+\frac{1}{2}y=1$
Parameterize (write a parametric equation for) each Cartesian equation by using $\text{\hspace{0.17em}}x\left(t\right)=a\mathrm{cos}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y(t)=b\mathrm{sin}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1.$
Parameterize the line from $\text{\hspace{0.17em}}(-2,3)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}(4,7)\text{\hspace{0.17em}}$ so that the line is at $\text{\hspace{0.17em}}(-2,3)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(4,7)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=1.$
$\{\begin{array}{l}x\left(t\right)=-2+6t\hfill \\ y\left(t\right)=3+4t\hfill \end{array}$
For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.
$\{\begin{array}{l}x\left(t\right)=3{t}^{2}\hfill \\ y\left(t\right)=2t-1\hfill \end{array}$
$\{\begin{array}{l}x(t)={e}^{t}\hfill \\ y(t)=-2{e}^{5\text{\hspace{0.17em}}t}\hfill \end{array}$
$y=-2{x}^{5}$
$\{\begin{array}{l}x(t)=3\mathrm{cos}\text{\hspace{0.17em}}t\hfill \\ y(t)=2\mathrm{sin}\text{\hspace{0.17em}}t\hfill \end{array}$
A ball is launched with an initial velocity of 80 feet per second at an angle of 40° to the horizontal. The ball is released at a height of 4 feet above the ground.
For the following exercises, determine whether the two vectors, $\text{\hspace{0.17em}}u\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}v,\text{\hspace{0.17em}}$ are equal, where $\text{\hspace{0.17em}}u\text{\hspace{0.17em}}$ has an initial point $\text{\hspace{0.17em}}{P}_{1}\text{\hspace{0.17em}}$ and a terminal point $\text{\hspace{0.17em}}{P}_{2},\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ has an initial point $\text{\hspace{0.17em}}{P}_{3}\text{\hspace{0.17em}}$ and a terminal point $\text{\hspace{0.17em}}{P}_{4}.$
${P}_{1}=\left(-1,4\right),{P}_{2}=\left(3,1\right),{P}_{3}=\left(5,5\right)$ and $\text{\hspace{0.17em}}{P}_{4}=\left(9,2\right)$
${P}_{1}=\left(6,11\right),{P}_{2}=\left(-2,8\right),{P}_{3}=\left(0,-1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{P}_{4}=\left(-8,2\right)$
not equal
For the following exercises, use the vectors $\text{\hspace{0.17em}}u=2i-j\text{,}v=4i-3j\text{,}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}w=-2i+5j\text{\hspace{0.17em}}$ to evaluate the expression.
For the following exercises, find a unit vector in the same direction as the given vector.
a = 8 i − 6 j
b = −3 i − j
$-\frac{3\sqrt{10}}{10}$ i $-\frac{\sqrt{10}}{10}$ j
For the following exercises, find the magnitude and direction of the vector.
$\u27e86,\mathrm{-2}\u27e9$
$\u27e8\mathrm{-3},\mathrm{-3}\u27e9$
Magnitude: $\text{\hspace{0.17em}}3\sqrt{2},\text{\hspace{0.17em}}$ Direction: $\text{225\xb0}$
For the following exercises, calculate $\text{\hspace{0.17em}}u\cdot v\text{.}$
u = −2 i + j and v = 3 i + 7 j
Given v $=\u3008\mathrm{-3},4\u3009$ draw v , 2 v , and $\text{\hspace{0.17em}}\frac{1}{2}$ v .
Given the vectors shown in [link] , sketch u + v , u − v and 3 v .
Given initial point $\text{\hspace{0.17em}}{P}_{1}=\left(3,2\right)\text{\hspace{0.17em}}$ and terminal point $\text{\hspace{0.17em}}{P}_{2}=\left(-5,-1\right),\text{\hspace{0.17em}}$ write the vector $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ in terms of $\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}j.\text{\hspace{0.17em}}$ Draw the points and the vector on the graph.
Assume $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\beta \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ Solve the triangle, if possible, and round each answer to the nearest tenth, given $\text{\hspace{0.17em}}\beta =\mathrm{68\xb0},b=21,c=16.$
$\alpha =\mathrm{67.1\xb0},\gamma =\mathrm{44.9\xb0},a=20.9$
Find the area of the triangle in [link] . Round each answer to the nearest tenth.
A pilot flies in a straight path for 2 hours. He then makes a course correction, heading 15° to the right of his original course, and flies 1 hour in the new direction. If he maintains a constant speed of 575 miles per hour, how far is he from his starting position?
$\text{1712miles}$
Convert $\text{\hspace{0.17em}}\left(2,2\right)\text{\hspace{0.17em}}$ to polar coordinates, and then plot the point.
Convert $\text{\hspace{0.17em}}\left(2,\frac{\pi}{3}\right)\text{\hspace{0.17em}}$ to rectangular coordinates.
$\left(1,\sqrt{3}\right)$
Convert the polar equation to a Cartesian equation: $\text{\hspace{0.17em}}{x}^{2}+{y}^{2}=5\mathrm{y.}$
Convert to rectangular form and graph: $r=-3\mathrm{csc}\text{\hspace{0.17em}}\theta .$
$y=-3$
Test the equation for symmetry: $\text{\hspace{0.17em}}r=-4\mathrm{sin}(2\theta ).$
Graph $\text{\hspace{0.17em}}r=3+3\mathrm{cos}\text{\hspace{0.17em}}\theta .$
Graph $\text{\hspace{0.17em}}r=3-5\text{sin}\text{\hspace{0.17em}}\theta .$
Find the absolute value of the complex number $5-9i.$
$\sqrt{106}$
Write the complex number in polar form: $\text{\hspace{0.17em}}4+i\text{.}$
Convert the complex number from polar to rectangular form: $\text{\hspace{0.17em}}z=5\text{cis}\left(\frac{2\pi}{3}\right).$
$\frac{-5}{2}+i\frac{5\sqrt{3}}{2}$
Given $\text{\hspace{0.17em}}{z}_{1}=8\mathrm{cis}\left(\mathrm{36\xb0}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{z}_{2}=2\mathrm{cis}\left(\mathrm{15\xb0}\right),$ evaluate each expression.
${z}_{1}{z}_{2}$
$\frac{{z}_{1}}{{z}_{2}}$
$4\mathrm{cis}\left(\mathrm{21\xb0}\right)$
${\left({z}_{2}\right)}^{3}$
$\sqrt{{z}_{1}}$
$2\sqrt{2}\mathrm{cis}\left(\mathrm{18\xb0}\right),2\sqrt{2}\mathrm{cis}\left(\mathrm{198\xb0}\right)$
Plot the complex number $\text{\hspace{0.17em}}\mathrm{-5}-i\text{\hspace{0.17em}}$ in the complex plane.
Eliminate the parameter $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ to rewrite the following parametric equations as a Cartesian equation: $\text{\hspace{0.17em}}\{\begin{array}{l}x(t)=t+1\hfill \\ y(t)=2{t}^{2}\hfill \end{array}.$
$y=2{\left(x-1\right)}^{2}$
Parameterize (write a parametric equation for) the following Cartesian equation by using $\text{\hspace{0.17em}}x\left(t\right)=a\mathrm{cos}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y(t)=b\mathrm{sin}\text{\hspace{0.17em}}t:$ $\frac{{x}^{2}}{36}+\frac{{y}^{2}}{100}=1.$
Graph the set of parametric equations and find the Cartesian equation: $\text{\hspace{0.17em}}\{\begin{array}{l}x(t)=-2\mathrm{sin}\text{\hspace{0.17em}}t\hfill \\ y(t)=5\mathrm{cos}\text{\hspace{0.17em}}t\hfill \end{array}.$
A ball is launched with an initial velocity of 95 feet per second at an angle of 52° to the horizontal. The ball is released at a height of 3.5 feet above the ground.
For the following exercises, use the vectors u = i − 3 j and v = 2 i + 3 j .
Calculate $\text{\hspace{0.17em}}u\cdot v.$
Find a unit vector in the same direction as $\text{\hspace{0.17em}}v.$
$\frac{2\sqrt{13}}{13}i+\frac{3\sqrt{13}}{13}j$
Given vector $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ has an initial point $\text{\hspace{0.17em}}{P}_{1}=\left(2,2\right)\text{\hspace{0.17em}}$ and terminal point $\text{\hspace{0.17em}}{P}_{2}=\left(-1,0\right),\text{\hspace{0.17em}}$ write the vector $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ in terms of $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}j.\text{\hspace{0.17em}}$ On the graph, draw $\text{\hspace{0.17em}}v,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}-v.\text{\hspace{0.17em}}$
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