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Use vector methods to prove the law of cosines

Let \theta denote the angle between two vectors A and B both of which are nonzero. Prove that

    \[ \lVert A - B \rVert^2 = \lVert A \rVert^2 + \lVert B \rVert^2 - 2 \lVert A \rVert \lVert B \rVert \cos \theta. \]


Proof. We compute

    \begin{align*}  \lVert A - B \rVert^2 &= (A-B) \cdot (A-B) \\  &= A \cdot A - 2A \cdot B + B \cdot B \\  &= \lVert A \rVert^2 + \lVert B \rVert^2 - 2 \left( \lVert A \rVert \lVert B \rVert \cos \theta). \qquad \blacksquare \end{align*}

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