Find tan theta

Question

If vec{v} times vec{w} = <2,5,4> and vec{v}•vec{w} = 2, find tan(theta), where theta is the angle between vec{v} and vec{w}.

 

tan(theta)=

 

Answer 1

v · w = |v| |w| cosθ

v × w = |v| |w| sinθ n

 

n is the unit vector perpendicular to both v and w. You can find n from the cross product given, and divide to get tanθ = sinθ / cosθ.

 

Divide the 2nd equation by the 1st, to get sinθ / cosθ on the right.

v × w / v · w = (|v| |w| sinθ n) / (|v| |w| cosθ)

⟨2, 5, 4⟩ / 2 = (sinθ / cosθ) n

⟨1, 5/2, 2⟩ = tanθ n

 

Since n is a unit vector, tanθ equals the magnitude of the vector on the left.