Let $\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ and let $\vec{b}$ be a vector such that $\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}$ and $\vec{a} \cdot \vec{b}=3$. Then the projection of $\vec{b}$ on the vector $\vec{a}-\vec{b}$ is :

<p>$\overrightarrow a = \widehat i - \widehat j + 2\widehat k$</p> <p>$\overrightarrow a \times \overrightarrow b = 2\widehat i - \widehat k$</p> <p>$\overrightarrow a \,.\,\overrightarrow b = 3$</p> <p>$$|\overrightarrow a \times \overrightarrow b {|^2} + |\overrightarrow a \,.\,\overrightarrow b {|^2} = |\overrightarrow a {|^2}\,.\,|\overrightarrow b {|^2}$$</p> <p>$\Rightarrow 5 + 9 = 6|\overrightarrow b {|^2}$</p> <p>$\Rightarrow |b {|^2} = {7 \over 3}$</p> <p>$$|\overrightarrow a - \overrightarrow b | = \sqrt {|\overrightarrow a {|^2} + |\overrightarrow b {|^2} - 2\overrightarrow a \,.\,\overrightarrow b } = \sqrt {{7 \over 3}} $$</p> <p>projection of $\overrightarrow b$ on $$\overrightarrow a - \overrightarrow b = {{\overrightarrow b \,.\,(\overrightarrow a - \overrightarrow b )} \over {|\overrightarrow a - \overrightarrow b |}}$$</p> <p>$$ = {{\overrightarrow b \,.\,\overrightarrow a - |\overrightarrow b {|^2}} \over {|\overrightarrow a - \overrightarrow b |}} = {{3 - {7 \over 3}} \over {\sqrt {{7 \over 3}} }}$$</p> <p>$= {2 \over {\sqrt {21} }}$</p>