Find theta from the cross product and dot product of two vectors

In summary, the cross product of vector v and vector w is equal to 3i + j + 4k, the dot product is equal to 4, and theta is the angle between the two vectors. To find tan(theta) and theta, the equations |3i + j + 4k| = |v|*|w|*sin(theta) and 4 = |v|*|w|*cos(theta) can be used. By isolating |v|*|w| in both equations and setting the resulting sides equal to each other, the equation tan(theta) = sqrt(26)/4 can be obtained. Therefore, theta is equal to arctan(sqrt(26)/4).
  • #1
loganblacke
48
0

Homework Statement


If the cross product of vector v cross vector w = 3i + j + 4k, and the dot product of vector v dot vector w = 4, and theta is the angle between vector v and vector w, find tan(theta) and theta.


Homework Equations



vector c = |v||w| sin(theta) where vector c is the cross product of v and w.

The Attempt at a Solution



I'm assuming you have to split the cross product back into the two original vectors and then calculate the angle but I'm not sure how to go from cross product to 2 vectors. Please help!
 
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  • #2
You can't get the two vectors. And you don't have to.
|3i + j + 4k|=|v|*|w|*sin(theta). 4=|v|*|w|*cos(theta). How would you get tan(theta) from that?
 
  • #3
Dick said:
You can't get the two vectors. And you don't have to.
|3i + j + 4k|=|v|*|w|*sin(theta). 4=|v|*|w|*cos(theta). How would you get tan(theta) from that?

I honestly have no idea.
 
  • #4
Think trig identity.
 
  • #5
vela said:
Think trig identity.

That's coy. :) What's the definition of tan(theta)?
 
  • #6
Dick said:
That's coy. :) What's the definition of tan(theta)?

tan theta is sin theta/cos theta.. which I think would put the vector over its magnitude and result in tan theta = unit vector..
 
  • #7
loganblacke said:
tan theta is sin theta/cos theta.. which I think would put the vector over its magnitude and result in tan theta = unit vector..

? Divide the two sides of the equations by each other. Can't you find a way to get tan(theta) on one side?
 
  • #8
Dick said:
? Divide the two sides of the equations by each other. Can't you find a way to get tan(theta) on one side?

I'm completely lost right now, the only thing i can work out on paper is if you isolate |v|*|w| in both equations by dividing both sides by cos theta and sin theta respectively. Then you could set the vector/sin theta = 4/cos theta.
 
  • #9
loganblacke said:
I'm completely lost right now, the only thing i can work out on paper is if you isolate |v|*|w| in both equations by dividing both sides by cos theta and sin theta respectively. Then you could set the vector/sin theta = 4/cos theta.

There aren't any vectors here anymore, there's only |3i + j + 4k|. That's number, not a vector. You can compute it. Can't you get sin(theta)/cos(theta) on one side and a number on the the other?
 
Last edited:
  • #10
Dick said:
There aren't any vectors here anymore. Everything is just numbers. Sure isolate |v|*|w| in both equations. Then set the other sides equal to each other. What's the resulting equation?

I see now that its the magnitude of vector 3i + J + 4k rather than the vector itself. So you end up with sqrt(3^2+1^2+4^2)/sin theta = 4/cos theta..

So you end up with tan theta = sqrt(26)/4.
 
  • #11
then theta = arctan(sqrt(26)/4)

Thanks for the help.. again.
 
  • #12
Dick said:
That's coy. :)
I am nothing if not coy. :wink:
 

1. What is the cross product of two vectors?

The cross product of two vectors is a vector that is perpendicular to both of the original vectors and has a magnitude equal to the product of the magnitudes of the two vectors multiplied by the sine of the angle between them.

2. What is the dot product of two vectors?

The dot product of two vectors is a scalar value that is equal to the product of the magnitudes of the two vectors multiplied by the cosine of the angle between them.

3. How do you find theta from the cross product and dot product of two vectors?

To find theta, the angle between two vectors, you can use the inverse cosine function on the result of the dot product divided by the product of the magnitudes of the two vectors. This will give you the cosine of the angle, which you can then use to find the angle itself.

4. Can theta be negative?

Yes, theta can be negative if the dot product of the two vectors is negative. This indicates that the angle between the two vectors is greater than 90 degrees.

5. What is the relationship between the cross product and dot product of two vectors?

The cross product and dot product of two vectors are related in that the dot product can be used to find the angle between the two vectors, which is then used in the formula for the cross product. Additionally, the dot product of two perpendicular vectors is always equal to 0, which means that the cross product of those two vectors will be a vector perpendicular to both of them.

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