JEE Main 2024 — 6 April, Shift 2 — Question 17

$$\vec{a}\times\left(\overset{{}}{\mathop{{\hat{i}}}}\,+\overset{{}}{\mathop{{\hat{j}}}}\, \right)=\left| \begin{matrix}{\hat{i}} & \overset{{}}{\mathop{{\hat{j}}}}\, & \overset{{}}{\mathop{{\hat{k}}}}\, \\2 & 1 & -1 \\1 & 1 & 0 \\\end{matrix} \right|$$

$=\hat{i}-\hat{j}+\hat{k}$

$(\vec{a} \times(\hat{i} \times \hat{j})) \times \hat{i}=\hat{k}+\hat{j}$ $((\vec{a} \times(\hat{i} \times \hat{j})) \times i) \times \hat{i}=\hat{j}-\hat{k}$

projection of $\vec{a}$

on $\hat{b}=\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

$=\frac{1+1}{\sqrt{2}}=\sqrt{2}$

the square of the projection of $\vec{a}$ on $\vec{b}$ is $2$