Exercise 3.5

]> Exercise 3.5

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Exercise 3.5

Write out $(\overset{⟶}{v} - \overset{⟶}{w}) \cdot (\overset{⟶}{v} - \overset{⟶}{w})$ 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 $(\overset{⟶}{v} - \overset{⟶}{w})$ , which is $(\overset{⟶}{v} - \overset{⟶}{w}, \overset{⟶}{v} - \overset{⟶}{w})$ can be expanded to $(\overset{⟶}{v}, \overset{⟶}{v}) + (\overset{⟶}{w}, \overset{⟶}{w}) - 2 (\overset{⟶}{v}, \overset{⟶}{w})$ which is the square of the length of $\overset{⟶}{v}$ plus the square of the length of $\overset{⟶}{w}$ minus twice the length of $\overset{⟶}{v}$ times the length of $\overset{⟶}{w}$ times the cosine of the angle between them. This is the law of cosines.