# Finding angle of rotation relating two vectors to a third

by pvm
Tags: angle, relating, rotation, vectors
 P: 692 I would approach it this way. I would define an orthogonal Cartesian coordinate system, lettinng h be along the z direction. Then unit vectors of my three axes, in terms of your vectors, could be something like this: $$\mathbf{\hat{z}} = \frac{\mathbf{h}}{ |\mathbf{h}|}.$$ $$\mathbf{\hat{y}} = \frac{\mathbf{h} \times \mathbf{v_1}}{|\mathbf{h} \times \mathbf{v_1}|}.$$ $$\mathbf{\hat{x}} = \mathbf{\hat{y}} \times \mathbf{\hat{z}}.$$ So that $\mathbf{v_1}$ only has x and z components; that is, the the projection of $\mathbf{v_1}$ onto the x-y plane coincides with the x axis. Now, since the rotation is about the z axis, we just need to project $\mathbf{v_2}$ onto the x-y plane (that is, find the x and y components) and determine the angle with repect to the x axis. I hope that helps. Jason