A) \[{{a}^{2}}-{{b}^{2}}\]
B) \[{{a}^{2}}+{{b}^{2}}\]
C) \[{{a}^{2}}{{b}^{2}}-{{(\mathbf{a}\,.\,\mathbf{b})}^{2}}\]
D) None of these
Correct Answer: C
Solution :
\[{{(\mathbf{a}\times \mathbf{b})}^{2}}=(\mathbf{a}\times \mathbf{b})\,.\,(\mathbf{a}\times \mathbf{b})=(ab\sin \theta \,\mathbf{\hat{n}})(ab\sin \theta \,\mathbf{\hat{n}})\] \[={{a}^{2}}{{b}^{2}}{{\sin }^{2}}\theta ={{a}^{2}}{{b}^{2}}(1-{{\cos }^{2}}\theta )\] \[={{a}^{2}}{{b}^{2}}-{{a}^{2}}{{b}^{2}}{{\cos }^{2}}\theta ={{a}^{2}}{{b}^{2}}-{{(\mathbf{a}\,.\,\mathbf{b})}^{2}}.\]
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