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

loganblacke

1. The problem statement, all variables and given/known data
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.


2. Relevant equations

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

3. 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!

Dick

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?

loganblacke

I honestly have no idea.

vela

Think trig identity.

Dick

That's coy. :) What's the definition of tan(theta)?

loganblacke

tan theta is sin theta/cos theta.. which I think would put the vector over its magnitude and result in tan theta = unit vector..

Dick

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

loganblacke

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.

Dick

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?

loganblacke

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.

loganblacke

then theta = arctan(sqrt(26)/4)

Thanks for the help.. again.

vela

I am nothing if not coy.