What is the cosine of the angle between the two unit vectors represented by (0.7071, 0.7071)T and (0.5, 0.866)T ?

cos θ  =  (0.7071, 0.7071)T · (0.5, 0.866)T   =   0.7071 · 0.5 + 0.7071 · 0.866   =   0.9659

# 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

### QUESTION 8:

What is the cosine of the angle between the two unit vectors represented by qu  =  (0.0, 1)T and ru  =  (0.5, 0.866)T ?