Let a=-i+2j+2k, b=8i+7j-3k and c be a vector such that a c= b.

Question

Let $\overset{	o }{a}=-\hat{i}+2\hat{j}+2\hat{k},\overset{	o \leftarrow }{b}=8\hat{i}+7\hat{j}-3\hat{k}$ and $\overset{	o }{c}$ be a vector such that $\overset{	o }{a}	imes \overset{	o }{c}=\overset{	o }{b}$. If $\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})=4$, then $|\vec{a}+\vec{c}{|}^{2}$ is equal to :

Accepted Answer

$\begin{gathered} \vec{c}=x\hat{i}+y\hat{j}+z\hat{k} \ \begin{gathered} \quad \vec{a}	imes \vec{c}=\vec{b}|\begin{array}{ccc}\hat{i} \quad \hat{j} \quad k \ -1 \quad 2 \quad 2 \ x \quad y \quad z\end{array}| \ =(2z-2y)\hat{i}+(z+2x)\hat{j}+(-y-2x)\hat{k} \ \equiv 8\hat{i}+7\hat{j}-3\hat{k} \end{gathered}$

$ \Rightarrow 2z-2y=8 \ z+2x=7 \ -y-2x=3y=3-2x,z=7-2x \end{gathered}$

$\begin{gathered} \vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})=4 \ x+y+z=4 \ x+3-2x+7-2x=4 \ \Rightarrow \, x=2 \ \, y=-1 \ \, z=3 \ \Rightarrow \, \vec{c}=2\hat{i}-\hat{j}+3\hat{k} \ |\vec{a}+\vec{c}{|}^{2}=|\hat{i}+\hat{j}+5\hat{k}{|}^{2} \ =27 \end{gathered}$

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