Law of Cosines

B) Proofs

dot product 를 활용한 증명

$c=\mathbf{a}-\mathbf{b}$

$$\begin{array}{rl}c\cdot c& =(\mathbf{a}-\mathbf{b})\cdot (\mathbf{a}-\mathbf{b})\\ & =\mathbf{a}\cdot \mathbf{a}-\mathbf{a}\cdot \mathbf{b}-\mathbf{b}\cdot \mathbf{a}+\mathbf{b}\cdot \mathbf{b}\\ & ={\mathbf{a}}^2-\mathbf{a}\cdot \mathbf{b}-\mathbf{a}\cdot \mathbf{b}+{\mathbf{b}}^2\\ & ={\mathbf{a}}^2-2\mathbf{a}\cdot \mathbf{b}+{\mathbf{b}}^2\\ c^2& ={\mathbf{a}}^2+{\mathbf{b}}^2-2\mathbf{ab}\cos \theta \end{array}$$

C) Related

D) References

• https://en.wikipedia.org/wiki/Law_of_cosines