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Exercise 3.5

Write out ( v ⟶ − w ⟶ ) · ( v ⟶ − w ⟶ ) using the linearity of the dot product in each of its arguments. What famous law does this establish?

Solution:

The square of the length of ( v ⟶ − w ⟶ ) , which is ( v ⟶ − w ⟶ , v ⟶ − w ⟶ ) can be expanded to ( v ⟶ , v ⟶ ) + ( w ⟶ , w ⟶ ) − 2 ( v ⟶ , w ⟶ ) which is the square of the length of v ⟶ plus the square of the length of w ⟶ minus twice the length of v ⟶ times the length of w ⟶ times the cosine of the angle between them. This is the law of cosines.

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