The direction of any vector A in three-dimensional space can be specified by giving the angles a, ß, and Ύ that the vector makes with the x, y, and z axes, respectively. If A = Axi + Ayj + Azk,
(a) Find expressions for cos a, cos ß, and cos Ύ (these are known as direction cosines), and (b) Show that these angles satisfy the relation cos2a + cos2ß + cos2Ύ = 1. (Hint: Take the scalar product of A with i, j, and k separately.)