Practice

The first vector, (0.7071, 0.7071)^T is a unit vector at 45°. The second vector, (0.5, 0.866)^T is a unit vector at 60°. So the dot product should be the cosine of 15°, which it is. Another way to see this is with symbols.

1. The first vector is ( cos 45, sin 45 )^T
2. The second vector is ( cos 60, sin 60 )^T
3. The dot product is cos 45 cos 60 + sin 45 sin 60
4.     =     cos 45 cos(45 + 15) + sin 45 sin(45+15)
5.     =     cos 45 (cos 45 cos 15 - sin 45 sin 15) +
   
          sin 45 (cos 45 sin 15 + cos 15 sin45)
6.     =     cos 45 cos 45 cos 15 - cos 45 sin 45 sin 15 +
   
          sin 45 cos 45 sin 15 + sin 45 cos 15 sin 45
7.     =     cos² 45 cos 15 + sin²45 cos 15
8.     =     (cos²45 + sin²45)cos15
9.     =     cos 15

Step 5 uses:

cos(x+y)  =  cos(x) cos(y) - sin(x) sin(y)
sin(x+y)  =  sin(x) cos(y) + cos(x) sin(y)

Step 9 uses:

sin²x + cos²x   =  1