A) \[\vec{c}-\left( \frac{\vec{a}.\,\vec{c}}{a.\,\vec{b}} \right)\vec{b}\]
B) \[\vec{b}-\left( \frac{\vec{b}.\,\vec{c}}{\vec{a}.\,\vec{b}} \right)\vec{c}\]
C) \[\vec{b}+\left( \frac{\vec{a}.\,\vec{c}}{\vec{a}.\,\vec{b}} \right)\vec{b}\]
D) \[\vec{b}+\left( \frac{\vec{b}.\,\vec{c}}{\vec{a}.\,\vec{b}} \right)\vec{c}\]
Correct Answer: A
Solution :
We have \[\vec{b}\times \vec{c}=\vec{b}\times \vec{d}\] \[\Rightarrow \,\,\vec{a}\times (\vec{b}\times \vec{c})=\vec{a}\times (\vec{b}\times \vec{d})\] \[\Rightarrow \,\,\,(\vec{a}.\,\vec{c})\vec{b}-(\vec{a}.\,\vec{b})\vec{c}=(\vec{a}.\,\vec{d})\vec{b}-(\vec{a}.\,\vec{b})\vec{d}\] \[\Rightarrow \,\,\,(\vec{a}.\,\vec{b})\vec{d}=-(\vec{a}.\,\vec{c})\vec{b}+(\vec{a}.\,\vec{b})\vec{c}\] \[\Rightarrow \,\,\,\vec{d}=-\frac{(\vec{a}.\,\vec{c})}{(\vec{a}.\,\vec{b})}\vec{b}+\vec{c}\]You need to login to perform this action.
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