Let a=-i+j+2k, b=i-j-3k, c= a b and d= c a.

Question

Let $\overset{	o }{a}=-\hat{i}+\hat{j}+2\hat{k},\overset{	o \leftarrow }{b}=\hat{i}-\hat{j}-3\hat{k},\overset{	o }{c}=\overset{	o }{a}	imes \overset{	o }{b}$ and $\overset{	o }{d}=\overset{	o }{c}	imes \overset{	o }{a}$. Then $(\vec{a}-\vec{b})\cdot \vec{d}$ is equal to :

Accepted Answer

$\begin{gathered} (\vec{a}-\vec{b})\cdot \vec{d}=\vec{a}\cdot \vec{d}-\vec{b}\cdot \vec{d} \ \because \vec{d}=\vec{c}	imes \vec{a}\Rightarrow \vec{a}\cdot \vec{d}=[\vec{a}\vec{c}\vec{a}]=0 \ \vec{b}\cdot \vec{d}=\vec{b}\cdot (\vec{c}	imes \vec{a}) \ =\vec{b}\cdot ((\vec{a}	imes \vec{b})	imes \vec{a}) \ \Rightarrow (\vec{a}-\vec{b})\cdot \vec{d}=\vec{b}\cdot (\vec{a}	imes (\vec{a}	imes \vec{b})) \ =\vec{b}\cdot [\vec{a}\cdot \vec{b}\vec{a}-\vec{a}\cdot \vec{a}\vec{b}] \ =(\vec{a}\cdot \vec{b}{)}^{2}-(\vec{a}\cdot \vec{a})(\vec{b}\cdot \vec{b}) \ =(-8{)}^{2}-6	imes 11=64-66=-2 \ \end{gathered}$

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