Could somebody explain how to do this:
Find the angle between the given vectors to the nearest tenth of a degree.
u = <-5, -4>, v = <-4, -3>
Answer choices:
A. -9.1 Degrees
B. 1.8 Degrees
C. 0.9 Degrees
D. 11.8 Degrees
Not sure how to find an angle between vectors.

- anonymous

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- schrodinger

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- anikhalder

Do you know how to find the magnitude or length of a vector?
And also so you know about scalar or dot product of 2 vectors?

- anonymous

I know Scalar and Dot Product. @anikhalder I sort of know magnitude...

- anikhalder

That's perfect! So let's go through this:
What is the length of u vector?

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- anonymous

I'm not sure how to solve for length...

- anikhalder

so its like, suppose i have a vector a = <3, 4>
Then the length of the vector a is :
\[\left| a \right|=\sqrt{3^{2} + {4^{2}}}\]
\[\left| a \right|=\sqrt{25}\]
That is the length of vector a is 5
So using the same way can you solve for the length of vector u?

- anonymous

Oh! Is the magnitude |u| = SqRt. 41 ?

- anikhalder

Perfect! You rock!
What about the magnitude (or length) of vector v?

- anonymous

|v| = 5 ?

- anikhalder

Awesome! Now do you know the formula for dot (or scalar) product of 2 vectors a and b?

- anonymous

Yes, would the product be: a x b = 32 ?

- anikhalder

Yes that's correct! But, I will suggest when writing dot product don't use 'x' sign. You will learn that there is another type of product called the vector or cross product which uses this sign...but we don't need to worry about right now.
Suppose I have a vector a and vector b
Then the dot product of vectors a and b is also defined as:
\[a.b=\left| a \right|\left| b \right|\cos \theta\]
Where theta is the angle between the vectors a and b.
So for our question at hand. Let's replace a by u and b by v.
So, we get:
\[u.v=\left| u\right|\left| v\right|\cos \theta\]
We can replace u.v as 32 and the magnitudes of u and v on the right hand side of the equation to get:
\[\cos \theta=\frac{ u.v }{ \left| u \right|\left| v \right|}\]
Can you solve for costheta now?

- anonymous

Is costheta = .99 ?

- anonymous

@anikhalder

- anikhalder

Yes! I find so as well. Now just find the cosine inverse of 0.99 and you'll get it in degrees

- anonymous

I got 8.1 degrees, approximately, but that is not one of my answers.

- anikhalder

That's what I get and now I must confess that I am confused as well

- anikhalder

Let's verify

- anonymous

Ok

- anikhalder

http://onlinemschool.com/math/assistance/vector/angl/
Can you type the values in and see what you get for costheta because I get 0.99 in this website as well

- anonymous

I am getting .99

- anonymous

@anikhalder Can you help me with another problem, and then maybe go back to this one?

- anikhalder

oh wait

- anonymous

Express the complex number in trigonometric form.
-3 + 3 square root of three i

- anikhalder

http://www.rapidtables.com/calc/math/Arccos_Calculator.htm

- anonymous

And I'm confused as to what that calculator meant.

- anikhalder

Check this, we were just using 0.99, we cant do that we have to use the full result which was 0.9995120761
plug this and find the cos inverse. You will see that we get our answer :)

- anonymous

Oh. I see. So the answer would be 1.8 degrees?

- anikhalder

Yep! And yes just take 4 minutes and see this video on complex numbers! he explains better than me :)))

- anonymous

What video?

- anikhalder

Sorry...my bad i forgot to give the link:
https://www.youtube.com/watch?v=6z6fzPXUbSQ

- anonymous

Thank you so much. (: I watched that video, though, and am still confused on my example. He helped me with a couple others, but the one I posted about is a bit confusing after watching the video.

- anikhalder

You mean its like :
\[3+3\sqrt{3}i\]
So,
\[\left| z \right| = \sqrt{3^{2}+(3\sqrt{3})^{2}} = 6\]
and \[\tan \theta = \frac{ 3\sqrt{3} }{ 3}\]
i.e. \[\theta = 60 degrees\]
Now, in polar or trigonometric form,
z = \[\left| z \right|(\cos \theta + i \sin \theta)\]
Just substitute the values and find the answer :)))

- anonymous

What is the Z value?

- anonymous

And is the theta value 120 degrees?

- anikhalder

you can say
\[\left| z \right|\] is like the absolute value of the complex number (like the|dw:1437768487456:dw|

- anonymous

So the z value is 3SqRt3. But you have in 3 as positive, the 3 is negative. So would that make the equation (in radians):
3SqRt2(Cos2pi/3 + i sin2pi/3)

- anikhalder

oho...i forgot the minus sign...yep!

- anonymous

Ok, so the theta value is 120 degrees? Not 150?

- anikhalder

Wait hold on, how is r coming to 3sqrt3 it should be 6

- anonymous

Oh. Ok, how do you get 6?

- anikhalder

|dw:1437768935393:dw|

- anonymous

Oh ok thanks. So it would be the same answer, but 6 as r vs. 3sqrt3 as r?

- anikhalder

yes it would 6(costheta + i sintheta)

- anonymous

Ok thanks! Theta = 2pi/3 right?

- anikhalder

yes. because tantheta in this case is 3sqrt3 / -3 which will give theta as 2pi/3

- anikhalder

:) I guess so. Please verify the answer with your book and your teacher. If I made any mistake please tell me :)))

- anonymous

Thanks so much!

- anikhalder

Thank you very much for bearing with me! My pleasure! Have a nice night! I will go to sleep!

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