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What angle is between a = (4,4,4)T and b = (4,0,4)T?


Visually, it looks somewhat less than midway between perpendicular and horizontal. 35° would be a good guess.

Formula for the angle between two Vectors

To do better than guessing, notice that in going from the tail to the head of a the vertical distance increases by 4 while the horizontal distance increases by 4 2.

Hence the tangent of the angle is 4 / (4 2)   =  1.0/2   =  0.7071.

so the angle with the horizontal is arctan( 0.7071 )  =  35.26°. Since b is in the horizontal plane, the angle between the two vectors must be that value.

The formula for the angle θ between two unit vectors is:

au · bu  =  cosθ

To use this formula with non-unit vectors:

  1. normalize each vector,
  2. compute the dot product,
  3. use the arc cos to get the angle.


(Calculator Problem: ) Apply this formula to a and b of the figure.