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Q.
Let $\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}$ 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 :-
Projection of $\vec{b}$ on vector $\vec{a}-\vec{b}$ is
$ =\frac{\vec{ b } \cdot(\vec{ a }-\vec{ b })}{|\vec{ a }-\vec{ b }|} $
$ =\frac{\vec{ a } \cdot \vec{ b }-| b |^2}{\sqrt{ a ^2+ b ^2-2 a \cdot b }}=\frac{3- b ^2}{\sqrt{6+ b ^2-6}}=\frac{3- b ^2}{ b }$
$ |\vec{ a } \times \vec{ b }|^2=5 $
$ a ^2 b ^2-( a \cdot b )^2=5$
$ 6 b ^2=14 \Rightarrow b ^2=\frac{7}{3} $
$ \therefore \frac{3- b ^2}{ b }=\frac{3-\frac{7}{3}}{\sqrt{\frac{7}{3}}}=2 \times \sqrt{21}$