Calculate the angle between: a = (10, 5)^T and b = (8, 12)^T

The lengths are: | a | = √125, | b | = √208.

The dot product is:

(10, 5)^T / √125 · (8, 12)^T / √208 = (10*8 + 5*12) / ( √125 √208)

= 140 / ( √125 √208) = 140 / √(125*208) = 140 / √26000 = 0.86824

so the angle is: arc cos ( 0.86824 ) = 29.745°

End of the Chapter

This chapter had quite a bit of numerical computation in it. Be careful that you did not loose sight of the important points:

dot product, affected by vector properties Two properties that are tangled together in the dot product.
angle, between unit vectors Formula for the angle between two unit vectors.
dot product, ambiguous orientation How many vectors have the same angular separation from a given vector?
cosine, angles which yield negative values What a negative cosine means for angular separation.
angle between non-unit vectors How to find the angle between non-unit vectors.