A good answer might be:

0.7071(1, 1)T ·  -0.7071(1, 1)T   =   0.7071(-0.7071)(1+1)
  =   -0.70712   *   2     =     -0.5*2     =   1

so the angle is: arc cos( -1 )   =  180°.     In 2D, there is only one vector 180° away from a given vector.


More Practice

Of course, the real world rarely gives you easy angles of 30°, 45°, 60°, and so on. Here is a more realistic problem: Find the angle between:

fu    =    (0.6, 0.8)T
gu    =    (0.8, 0.6)T

You may wish to confirm that these are indeed unit vectors. Sketch the vectors. Then, calculate their dot product. Use the arc cos function of the desktop calculator (or a real calculator) to find the angle. Calculate the answer in degrees, not radians.


QUESTION 13:

(Calculator Problem: )What is the angle?