The value of the cross product \(\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right)\) of two vectors \(\overrightarrow a - \overrightarrow b\) and \(\overrightarrow a + \overrightarrow b \) is:

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UP TGT Mathematics 2013 Official Paper

Answer 1

Calculation:

Given, \(( \vec a- \vec b) \times (\vec a + \vec b)\)

\(\Rightarrow (\vec a \times \vec a) + ( \vec a \times \vec b) - ( \vec b \times \vec a ) - ( \vec b \times \vec b)\)

we know that, \(\vec a \times \vec a = 0\;, \vec b \times \vec b = 0\; and, \vec a \times \vec b = - (\vec b \times \vec a)\)

\(\Rightarrow (\vec a \times \vec b) - (\vec b \times \vec a)\)

\(\Rightarrow (\vec a \times \vec b) -[ -(\vec a \times \vec b)]\)

\(= ( \vec a \times \vec b) + ( \vec a \times \vec b) = 2 (\vec a \times \vec b)\)