Understand that the picture they have drawn is in three dimensions. Visualize the x-axis running west-east on your page, the y-axis running north-south, and the z-axis rising up like a pole from your table top.
Again, this problem is just an
application of the cosine formula. You will need the three vectors in the direction of the x-, y-, and z-axes.
Note, on any given axis, the other two coordinates are 0. For example, on the x-axis, both y and z are 0. So, the easiest vector to use in the direction of the x-axis is (1,0,0). You could use anything in the form (k,0,0) as long as k is not 0. Thus,
alpha is the angle between the given vector (2,3,4) and the vector (1,0,0).