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# The Parallelogram Law states that $${\mid a + b \mid}^2 + {\mid a - b \mid}^2 = 2 {\mid a \mid}^2 + 2 {\mid b \mid}^2$$(a) Give a geometric interpretation of the Parallelogram Law.(b) Prove the Parallelogram Law. (See the hint in Exercise 62.)

## a) The sum of the square of the length of the diagonals of a parallelogram, are equal to the square of the length of the four sides of the parallelogram.b) $|\mathbf{a}+\mathbf{b}|^{2}+|\mathbf{a}-\mathbf{b}|^{2}=(\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}+\mathbf{b})+(\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}+\mathbf{b})$$=2(\mathbf{a} \cdot \mathbf{a})+2(\mathbf{b} \cdot \mathbf{b})$$=2|\mathbf{a}|^{2}+2|\mathbf{b}|^{2}$$=L . H . S$$=R . H . S \quad$ (R.T.P)

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