Question

$\overrightarrow{a} = \widehat{i} - 2\widehat{j} + 3\widehat{k},\overrightarrow{b} = 2\widehat{i} + \widehat{j} - \widehat{k},\overrightarrow{c} = \lambda\widehat{i} + \widehat{j} + \widehat{k}$, and $\overrightarrow{v} = \overrightarrow{a} \times \overrightarrow{b}$. If $\overrightarrow{v} \cdot \overrightarrow{c} = 11$

$${\overrightarrow{v} = (\overrightarrow{a} \times \overrightarrow{b}) = ( - \widehat{i} + 7\widehat{j} + 5\widehat{k}) }{\overrightarrow{v} \cdot \overrightarrow{c} = 11 = ( - \widehat{i} + 7\widehat{j} + 5\widehat{k}) \cdot (\lambda\widehat{i} + \widehat{j} + \widehat{k}) = 11 }{\Rightarrow - \lambda + 7 + 5 = 11 }{\Rightarrow \lambda = 1 }$$Length of projection of $\overrightarrow{b}$ on $\overrightarrow{c} = \overrightarrow{b} \cdot \widehat{c}$

$${\Rightarrow \left| (2\widehat{i} + \widehat{j} - \widehat{k}) \cdot \frac{(\widehat{i} + \widehat{j} + \widehat{k})}{\sqrt{3}} \right| = \frac{2 + 1 - 1}{\sqrt{3}} = P = \frac{2}{\sqrt{3}} }{\Rightarrow 9p^{2} = 9\left( \frac{4}{3} \right) = 12}$$