Let \(\vec{a}=\overset{^}{i}−2\overset{^}{j}+3\overset{^}{k},\vec{b}=2\overset{^}{i}+\overset{^}{j}−\overset{^}{k},\vec{…

Given: \(\vec{a}=\overset{^}{i}−2\overset{^}{j}+3\overset{^}{k},\vec{b}=2\overset{^}{i}+\overset{^}{j}−\overset{^}{k},\vec{c}=\lambda \overset{^}{i}+\overset{^}{j}+\overset{^}{k}\)

And, \(\vec{v}=\vec{a}\times \vec{b}\)

Then, \(\vec{v}=(\vec{a} \times \vec{b})=(-\hat{i}+7 \hat{j}+5 \hat{k})\)

Now, \(\vec{v} \cdot \vec{c}=11\)

\(\Rightarrow (-\hat{i}+7 \hat{j}+5 \hat{k}) \cdot(\lambda \hat{i}+\hat{j}+\hat{k})=11\)

\(\Rightarrow −\lambda +7+5=11\)

\(\Rightarrow \lambda =1\)

Length of projection of \(\vec{b}\) on \(\vec{c}=\vec{b}⋅\overset{^}{c}\)

\(\Rightarrow\left|(2 \hat{i}+\hat{j}-\hat{k}) \cdot \frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}\right|=\frac{2+1-1}{\sqrt{3}}\)

\(p=\frac{2}{\sqrt{3}}\)

\(\Rightarrow 9{p}^{2}=9\left(\frac{4}{3}\right)=12\)