Let a=i-3j+7k, b=2i-j+k and c be a vector such that ( a+2 b) c=3( c a).

Question

Let $\overset{	o }{a}=\hat{i}-3\hat{j}+7\hat{k},\overset{	o }{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\overset{	o }{c}$ be a vector such that $(\overset{	o }{a}+2\overset{	o }{b})	imes \overset{	o }{c}=3(\overset{	o }{c}	imes \overset{	o }{a})$.
If $\vec{a}\cdot \vec{c}=130$, then $\vec{b}\cdot \vec{c}$ is equal to \_\_\_\_\_\_\_\_\_\_.

Answer: 30

Accepted Answer

$(\vec{a}+2\vec{b})	imes \vec{c}=3(\vec{c}	imes \vec{a})$

$\begin{gathered} \Rightarrow \, \vec{b}	imes \vec{c}+2(\vec{a}	imes \vec{c})=0 \ (\vec{b}+2\vec{a})	imes \vec{c}=0 \ \vec{c}=\lambda (\vec{b}+2\vec{a}) \ \vec{c}\cdot \vec{a}=130\Rightarrow \lambda =1 \ \vec{c}=4\hat{i}-7\hat{j}+15\hat{k} \ \vec{b}.\vec{c}=30 \end{gathered}$

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