a = i + 2j - 3k, b = 3i - j + 2k, show that (a +b ) and (a -b ) a… - Vidyadip

Question

$\overrightarrow{a} = \hat{i} + 2\hat{j} - 3\hat{k}, \overrightarrow{b} = 3\hat{i} - \hat{j} + 2\hat{k}, \text{show that}\bigg(\overrightarrow{a} +\overrightarrow{b}\bigg) \text{and} \bigg(\overrightarrow{a} -\overrightarrow{b}\bigg)$ are perpendicular to each other.

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Answer

$(\overrightarrow{a} +\overrightarrow{b}) = 4\hat{i} + \hat{j} - \hat{k} ,\ (\overrightarrow{a} -\overrightarrow{b}) = - 2\hat{i} + 3\hat{j} -5\hat{k}$

$\text{For} (\overrightarrow{a} +\overrightarrow{b})\ and\ (\overrightarrow{a} -\overrightarrow{b}) \text{ to be perpendicular } (\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a} -\overrightarrow{b})$

$(\overrightarrow{a} +\overrightarrow{b}).(\overrightarrow{a}-\overrightarrow{b}) = 4(-2) + 1.3 + (-1)(-5)$

$= - 8 + 3 + 5 = 0$

$\therefore (\overrightarrow{a}+\overrightarrow{b}) \perp (\overrightarrow{a}-\overrightarrow{b})$

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