For any two vectors a and b , (a b)^2 is equal to - Vidyadip

MCQ

For any two vectors $a $ and $b$ , ${(a \times b)^2}$ is equal to

A
${a^2} - {b^2}$
B
${a^2} + {b^2}$
✓
${a^2}{b^2} - {(a\,.\,b)^2}$
D
None of these

✓

Answer

Correct option: C.

${a^2}{b^2} - {(a\,.\,b)^2}$

c
(c) ${(a \times b)^2} = (a \times b)\,.\,(a \times b) = (ab\sin \theta \,\hat n)(ab\sin \theta \,\hat n)$

$ = {a^2}{b^2}{\sin ^2}\theta = {a^2}{b^2}(1 - {\cos ^2}\theta )$

$ = {a^2}{b^2} - {a^2}{b^2}{\cos ^2}\theta = {a^2}{b^2} - {(a\,.\,b)^2}.$

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