# Thread: Proof of angle between vectors (using dot and cross product)

1. ## Proof of angle between vectors (using dot and cross product)

Prove that if $\theta$ is the angle between u and v and u . v =/= 0, then tan $\theta$ = ||u x v||/(u . v)

where . is dot product

I am pretty familiar with the properties of both dot and cross product but that tan throws me for a loop, I just can't seem to find a way to do this

2. Originally Posted by Idlewild
Prove that if [Theta] is the angle between u and v and u . v =/= 0, then tan [Theta] = ||u x v||/(u . v)

where . is dot product

I am pretty familiar with the properties of both dot and cross product but that tan throws me for a loop, I just can't seem to find a way to do this
Recall that $\displaystyle \sin(\theta)=\frac{\|a\times b\|}{\|a\|\|b\|}$ and $\displaystyle \cos(\theta)=\frac{a\cdot b}{\|a\|\|b\|}$.

3. Thanks for the speedy reply!

I'm not sure if this is too basic, or only physics related or something but I guess that means to use SOH CAH TOA, where O = || a x b ||, and A = a . b

Therefore tan $\theta$ = || a x b || / a . b
and ah! that turns out to be the given answer!

now to prove it... I see that u . v =/= 0, which means that u and v are not orthogonal.. That's all I really can come to right now, I haven't had much luck with the geometric interpretation

Edit: Or am I complicating this, and simply showing the explanation of sin and cos to tan would be sufficient?

4. You mean saying $\displaystyle\tan\theta = \frac{\sin \theta}{\cos \theta} =\frac{\frac{\|a\times b\|}{\|a\|\|b\|}}{\frac{a\cdot b}{\|a\|\|b\|}} = \frac{\|a\times b\|}{a\cdot b}$ ?

5. Yes, I was going to show the SOH CAH TOA thing, but that works.
Do you think that would be acceptable?

6. Maybe it would be acceptable, depends on your lecturer, you may need to prove

$\displaystyle \sin(\theta)=\frac{\|a\times b\|}{\|a\|\|b\|}$ and $\displaystyle \cos(\theta)=\frac{a\cdot b}{\|a\|\|b\|}$

as well for the answer to be accepted, this brings you back to square one.

7. Originally Posted by pickslides
Maybe it would be acceptable, depends on your lecturer, you may need to prove

$\displaystyle \sin(\theta)=\frac{\|a\times b\|}{\|a\|\|b\|}$ and $\displaystyle \cos(\theta)=\frac{a\cdot b}{\|a\|\|b\|}$

as well for the answer to be accepted, this brings you back to square one.
You don't really "prove" these though. By definition one takes $\displaystyle \cos(\theta)=\frac{a\cdot b}{\|a\|\|b\|}$ and the other one is gotten from the identity $\sin^2(\theta)+\cos^2(\theta)=1$.

8. It is the first part of the first question on this assignment and I do also think that we are given the definition for the cos, so I will just show by identities

Thanks everyone

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# proving the cross product in relation to sine theta

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