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WZ
Numerade Educator

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Problem 63 Hard Difficulty

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.)

Answer

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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Video Transcript

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