4.7 Vectors  (Page 11/22)

Evaluate the cube root of $z$ when $z=64\mathrm{cis}\left(\mathrm{210°}\right).$

Evaluate the square root of $z$ when $z=25\mathrm{cis}\left(\frac{3\pi }{2}\right).$

$6-2i$

$-1+3i$

$\{\begin{array}{l}x\left(t\right)=3t-1\hfill \\ y\left(t\right)=\sqrt{t}\hfill \end{array}$

$\{\begin{array}{l}x(t)=-\cos t\hfill \\ y(t)=2{\sin }^{2}t \hfill \end{array}$

Parameterize (write a parametric equation for) each Cartesian equation by using $x\left(t\right)=a\cos t$ and $y(t)=b\sin t$ for $\frac{x^2}{25}+\frac{y^2}{16}=1.$

Parameterize the line from $(-2,3)$ to $(4,7)$ so that the line is at $(-2,3)$ at $t=0$ and $(4,7)$ at $t=1.$

$\{\begin{array}{l}x\left(t\right)=3t^2\hfill \\ y\left(t\right)=2t-1\hfill \end{array}$

$\{\begin{array}{l}x(t)=e^t\hfill \\ y(t)=-2e^{5t}\hfill \end{array}$

$\{\begin{array}{l}x(t)=3\cos t\hfill \\ y(t)=2\sin t\hfill \end{array}$

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.

1. Find the parametric equations to model the path of the ball.
2. Where is the ball after 3 seconds?
3. How long is the ball in the air?

$P_1=\left(-1,4\right),P_2=\left(3,1\right),P_3=\left(5,5\right)$ and $P_4=\left(9,2\right)$

$P_1=\left(6,11\right),P_2=\left(-2,8\right),P_3=\left(0,-1\right)$ and $P_4=\left(-8,2\right)$

u − v

2 v − u + w

a = 8 i − 6 j

b = −3 i − j

$⟨6,\mathrm{-2}⟩$

$⟨\mathrm{-3},\mathrm{-3}⟩$

u = −2 i + j and v = 3 i + 7 j

u = i + 4 j and v = 4 i + 3 j

Given v $=〈\mathrm{-3},4〉$ draw v , 2 v , and $\frac{1}{2}$ v .

Given the vectors shown in [link] , sketch u + v , u − v and 3 v .

Given initial point $P_1=\left(3,2\right)$ and terminal point $P_2=\left(-5,-1\right),$ write the vector $v$ in terms of $i$ and $j.$ Draw the points and the vector on the graph.

Assume $\alpha$ is opposite side $a,\beta$ is opposite side $b,$ and $\gamma$ is opposite side $c.$ Solve the triangle, if possible, and round each answer to the nearest tenth, given $\beta =\mathrm{68°},b=21,c=16.$

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?

Convert $\left(2,2\right)$ to polar coordinates, and then plot the point.

Convert $\left(2,\frac{\pi }{3}\right)$ to rectangular coordinates.

Convert the polar equation to a Cartesian equation: $x^2+y^2=5\mathrm{y.}$

Convert to rectangular form and graph: $r=-3\csc \theta .$

Test the equation for symmetry: $r=-4\sin (2\theta ).$

Graph $r=3+3\cos \theta .$

Graph $r=3-5\text{sin}\theta .$

Find the absolute value of the complex number $5-9i.$

Write the complex number in polar form: $4+i\text{.}$

Convert the complex number from polar to rectangular form: $z=5\text{cis}\left(\frac{2\pi }{3}\right).$

$z_1{z}_2$

$\frac{z_1}{z_2}$

${\left(z_2\right)}^3$

$\sqrt{z_1}$

Plot the complex number $\mathrm{-5}-i$ in the complex plane.

Eliminate the parameter $t$ to rewrite the following parametric equations as a Cartesian equation: $\{\begin{array}{l}x(t)=t+1\hfill \\ y(t)=2t^2\hfill \end{array}.$

Parameterize (write a parametric equation for) the following Cartesian equation by using $x\left(t\right)=a\cos t$ and $y(t)=b\sin t:$ $\frac{x^2}{36}+\frac{y^2}{100}=1.$

Graph the set of parametric equations and find the Cartesian equation: $\{\begin{array}{l}x(t)=-2\sin t\hfill \\ y(t)=5\cos t\hfill \end{array}.$

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.

1. Find the parametric equations to model the path of the ball.
2. Where is the ball after 2 seconds?
3. How long is the ball in the air?

Find 2 u − 3 v .

Calculate $u\cdot v.$

Find a unit vector in the same direction as $v.$

Given vector $v$ has an initial point $P_1=\left(2,2\right)$ and terminal point $P_2=\left(-1,0\right),$ write the vector $v$ in terms of $i$ and $j.$ On the graph, draw $v,$ and $-v.$