I recently came across this math problem that seems impossible but I feel like it should have a solution.
Given three unit vectors a, b, and c, require that a x (b x c) = 1/2 b. Find the angles a makes with b and c.
So far, I've thought up three possible approaches to this problem. 1. Brute force calculation of the triple product using dummy variables for the vector components which should lead to some system of equations. Using the determinant definition of the vector cross product. 2. Using the geometric definition of the vector cross product to derive equations for the angles between them. 3. Some kind of analysis of the triple product that interprets the problem statement. b x c is a normal to the b-c plane, also a x (b x c) is a normal to the a-(b x c) plane. Since a x (b x c) is a scalar multiple of b, there could be an intuitive relationship between the three.
I've followed the three approaches to no avail. It seems I'm missing something. To clarify, I understand how to get angles between vectors using the dot product and how to find the vector triple product on its own.
Is there a way to solve this, using either of my approaches or a novel one?