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

$\begin{array}{r}\overrightarrow{a}=\widehat{i}-\widehat{j}+2\widehat{k}\\ \overrightarrow{a}\times \overrightarrow{b}=2\widehat{i}-\widehat{k}\\ \overrightarrow{a}\cdot \overrightarrow{b}=3\end{array}$
$\begin{array}{l}|\overrightarrow{a}\times \overrightarrow{b}{|}^2+|\overrightarrow{a}\cdot \overrightarrow{b}{|}^2=|\overrightarrow{a}{|}^2\cdot |\overrightarrow{b}{|}^2\\ \Rightarrow 5+9=6|\overrightarrow{b}{|}^2\\ \Rightarrow |b{|}^2=\frac{7}{3}\end{array}$
$|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2-2\overrightarrow{a}\cdot \overrightarrow{b}}=\sqrt{\frac{7}{3}}$
$\text{ projection of }\overrightarrow{b}\text{ on }\overrightarrow{a}-\overrightarrow{b}=\frac{\overrightarrow{b}\cdot (\overrightarrow{a}-\overrightarrow{b})}{|\overrightarrow{a}-\overrightarrow{b}|}$
$\begin{array}{l}=\frac{\overrightarrow{b}\cdot \overrightarrow{a}-|\overrightarrow{b}{|}^2}{|\overrightarrow{a}-\overrightarrow{b}|}=\frac{3-\frac{7}{3}}{\sqrt{\frac{7}{3}}}\\ =\frac{2}{\sqrt{21}}\end{array}$