What is the direction of the vector B?

Question

A vector A points 14° from the x axis. Vector B is twice as long as A. Their product A×B has length A^2 and points in the negative z direction. What is the direction of the vector B?

Answer 1

|cross(A,B)| = |A|*|B|*sin(theta)

|A^2| = |A|*2|A|*sin(theta)

|A^2| = 2*|A^2|*sin(theta)

...

sin(theta) = 1/2 so theta = 30 degrees
so there is 30 degrees between A and B so B is either 14+30 or 14-30.To figure this out, consider the direction of the cross product. A is directed 14 degrees from the x-axis, assuming above it for a positive angle. AxB is said to have a direction along the negative z axis. This means that B must be below A so that AxB is pointed along the negative z axis by using the right hand rule.So I deduce that the direction of B is 14-30 from the x axis, or -16 degrees from the x axis.

Answer 2

|R| = |A|•|B|sinθ
a^2 = - (A)(2A)sinθ
sinθ = - 1/2
θ = - 30°
14° - 30° = - 16°
- 16° from the x-axis