QJEEM25Ma1b18: JEE Main 2025 Physics - Class 11 JEEM25Ma1b18 Solution: Let and . Then the projection of on is :

$\begin{aligned}&\begin{aligned}& \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{a}} \times(\hat{\mathrm{i}}-3 \hat{\mathrm{k}}) \\& =\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\3 & -1 & 2 \\1 & 0 & -2\end{array}\right|=2 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}+\hat{\mathrm{k}} \\& \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \hat{\mathrm{k}}=8 \hat{\mathrm{i}}-2 \hat{\mathrm{j}} \\& \overrightarrow{\mathrm{c}}-2 \hat{\mathrm{j}}=8 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}\end{aligned}\\&\text { Projection of }(\hat{i}-2 \hat{j}) \text { on } \vec{a}\\&\begin{aligned}& (\overrightarrow{\mathrm{c}}-2 \hat{\mathrm{j}}) \cdot \hat{\mathrm{a}}=\frac{\langle 8,-4,0\rangle \cdot\langle 3,-1,2\rangle}{\sqrt{14}} \\& =\frac{28}{\sqrt{14}}=2 \sqrt{14}\end{aligned}\end{aligned}$