Let a=3i+j-2k,b=4i+j+7k and c=i-3j+4k be three vectors.

Question

Let $\vec{a}=3\hat{i}+\hat{j}-2\hat{k},\vec{b}=4\hat{i}+\hat{j}+7\hat{k}$ and $\vec{c}=\hat{i}-3\hat{j}+4\hat{k}$ be three vectors. If a vectors $\vec{p}$ satisfies $\vec{p}	imes \vec{b}=\vec{c}	imes \vec{b}$ and $\vec{p}\cdot \vec{a}=0$, then $\vec{p}\cdot (\hat{i}-\hat{j}-\hat{k})$ is equal to

Accepted Answer

$\begin{gathered} \overset{	o }{p}	imes \overset{	o }{b}-\overset{	o }{c}	imes \overset{	o }{b}=\overset{	o }{0} \ (\overset{	o }{p}-\overset{	o }{c})	imes \overset{	o }{b}=\overset{	o }{0} \ \overset{	o }{p}-\overset{	o }{c}=\lambda \overset{	o }{b}\Rightarrow \overset{	o }{p}=\overset{	o }{c}+\lambda \overset{	o }{b} \end{gathered}$

Now, $\overset{	o }{p}\cdot \overset{	o }{a}=0$ (given)

$\begin{gathered} \overset{	o }{c}\cdot \overset{	o }{a}+\lambda \overset{	o }{a}\cdot \overset{	o }{b}=0 \ (3-3-8)+\lambda (12+1-14)=0 \ \lambda =-8 \ \overset{	o }{p}=\overset{	o }{c}-8\overset{	o }{b} \ \overset{	o }{p}=-31\hat{i}-11\hat{j}-52\hat{k} \ \overset{	o }{p}\cdot (\hat{i}-\hat{j}-\hat{k}) \ =-31+11+52 \ =32 \end{gathered}$

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