]> Exercise 3.5

## 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.