A) \[\frac{(\vec{a}\times \vec{b})\cdot \vec{b}}{{{\left| {\vec{b}} \right|}^{2}}}\]
B) \[\frac{(\vec{a}\cdot \vec{b})\vec{a}}{{{\left| {\vec{a}} \right|}^{2}}}\]
C) \[\frac{(\vec{a}\cdot \vec{b})(\vec{b}\times \vec{a})}{{{\left| {\vec{b}} \right|}^{2}}}\]
D) \[\frac{(\vec{a}\cdot \vec{b})(\vec{b}\times \vec{a})}{\left| \vec{b}\times \vec{a} \right|}\]
Correct Answer: C
Solution :
[c] \[{{\vec{a}}_{1}}=(\vec{a}\cdot \hat{b})\hat{b}=\frac{(\vec{a}\cdot \vec{b})\vec{b}}{|\vec{b}{{|}^{2}}}\] \[\Rightarrow {{\vec{a}}_{2}}=\vec{a}-{{\vec{a}}_{1}}=\vec{a}-\frac{(\vec{a}\cdot \vec{b})\vec{b}}{|\vec{b}{{|}^{2}}}\] Thus, \[{{\vec{a}}_{1}}\times {{\vec{a}}_{2}}=\frac{(\vec{a}\cdot \vec{b})\vec{b}}{|\vec{b}{{|}^{2}}}\times \left( \vec{a}-\frac{(\vec{a}\cdot \vec{b})\vec{b}}{|\vec{b}{{|}^{2}}} \right)\] \[=\frac{(\vec{a}\cdot \vec{b})(\vec{b}\times a)}{|\vec{b}{{|}^{2}}}\]
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