# 4.7 Vectors  (Page 10/22)

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Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30° from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.

5.1583 pounds, 75.8° from the horizontal

## Non-right Triangles: Law of Sines

For the following exercises, 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 each triangle, if possible. Round each answer to the nearest tenth.

$\beta =50°,a=105,b=45$

Not possible

$\alpha =43.1°,a=184.2,b=242.8$

Solve the triangle.

$C=120°,a=23.1,c=34.1$

Find the area of the triangle.

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25° and 49°, as shown in [link] . Find the distance of the plane from point $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and the elevation of the plane.

distance of the plane from point $\text{\hspace{0.17em}}A:\text{\hspace{0.17em}}$ 2.2 km, elevation of the plane: 1.6 km

## Non-right Triangles: Law of Cosines

Solve the triangle, rounding to the nearest tenth, assuming $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ s opposite side

Solve the triangle in [link] , rounding to the nearest tenth.

$B=71.0°,C=55.0°,a=12.8$

Find the area of a triangle with sides of length 8.3, 6.6, and 9.1.

To find the distance between two cities, a satellite calculates the distances and angle shown in [link] (not to scale). Find the distance between the cities. Round answers to the nearest tenth.

40.6 km

## Polar Coordinates

Plot the point with polar coordinates $\text{\hspace{0.17em}}\left(3,\frac{\pi }{6}\right).$

Plot the point with polar coordinates $\text{\hspace{0.17em}}\left(5,-\frac{2\pi }{3}\right)$

Convert $\text{\hspace{0.17em}}\left(6,-\frac{3\pi }{4}\right)\text{\hspace{0.17em}}$ to rectangular coordinates.

Convert $\text{\hspace{0.17em}}\left(-2,\frac{3\pi }{2}\right)\text{\hspace{0.17em}}$ to rectangular coordinates.

$\text{\hspace{0.17em}}\left(0,2\right)\text{\hspace{0.17em}}$

Convert $\left(7,-2\right)$ to polar coordinates.

Convert $\left(-9,-4\right)$ to polar coordinates.

$\left(9.8489,203.96°\right)$

For the following exercises, convert the given Cartesian equation to a polar equation.

$x=-2$

${x}^{2}+{y}^{2}=64$

$r=8$

${x}^{2}+{y}^{2}=-2y$

For the following exercises, convert the given polar equation to a Cartesian equation.

$r=7\text{cos}\text{\hspace{0.17em}}\theta$

${x}^{2}+{y}^{2}=7x$

$r=\frac{-2}{4\mathrm{cos}\text{\hspace{0.17em}}\theta +\mathrm{sin}\text{\hspace{0.17em}}\theta }$

For the following exercises, convert to rectangular form and graph.

$\theta =\frac{3\pi }{4}$

$y=-x$

$r=5\mathrm{sec}\text{\hspace{0.17em}}\theta$

## Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

$r=4+4\mathrm{sin}\text{\hspace{0.17em}}\theta$

symmetric with respect to the line $\theta =\frac{\pi }{2}$

$r=7$

Sketch a graph of the polar equation $\text{\hspace{0.17em}}r=1-5\mathrm{sin}\text{\hspace{0.17em}}\theta .\text{\hspace{0.17em}}$ Label the axis intercepts.

Sketch a graph of the polar equation $\text{\hspace{0.17em}}r=5\mathrm{sin}\left(7\theta \right).$

Sketch a graph of the polar equation $\text{\hspace{0.17em}}r=3-3\mathrm{cos}\text{\hspace{0.17em}}\theta$

## Polar Form of Complex Numbers

For the following exercises, find the absolute value of each complex number.

$-2+6i$

$4-\text{​}3i$

5

Write the complex number in polar form.

$5+9i$

$\frac{1}{2}-\frac{\sqrt{3}}{2}\text{​}i$

$\mathrm{cis}\left(-\frac{\pi }{3}\right)$

For the following exercises, convert the complex number from polar to rectangular form.

$z=5\mathrm{cis}\left(\frac{5\pi }{6}\right)$

$z=3\mathrm{cis}\left(40°\right)$

$2.3+1.9i$

For the following exercises, find the product $\text{\hspace{0.17em}}{z}_{1}{z}_{2}\text{\hspace{0.17em}}$ in polar form.

${z}_{1}=2\mathrm{cis}\left(89°\right)$

${z}_{2}=5\mathrm{cis}\left(23°\right)$

${z}_{1}=10\mathrm{cis}\left(\frac{\pi }{6}\right)$

${z}_{2}=6\mathrm{cis}\left(\frac{\pi }{3}\right)$

$60\mathrm{cis}\left(\frac{\pi }{2}\right)$

For the following exercises, find the quotient $\text{\hspace{0.17em}}\frac{{z}_{1}}{{z}_{2}}\text{\hspace{0.17em}}$ in polar form.

${z}_{1}=12\mathrm{cis}\left(55°\right)$

${z}_{2}=3\mathrm{cis}\left(18°\right)$

${z}_{1}=27\mathrm{cis}\left(\frac{5\pi }{3}\right)$

${z}_{2}=9\mathrm{cis}\left(\frac{\pi }{3}\right)$

$3\mathrm{cis}\left(\frac{4\pi }{3}\right)$

For the following exercises, find the powers of each complex number in polar form.

Find $\text{\hspace{0.17em}}{z}^{4}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=2\mathrm{cis}\left(70°\right)$

Find $\text{\hspace{0.17em}}{z}^{2}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=5\mathrm{cis}\left(\frac{3\pi }{4}\right)$

$25\mathrm{cis}\left(\frac{3\pi }{2}\right)$

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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