QJEEM25Ma683e: JEE Main 2025 Physics - Class 11 JEEM25Ma683e Solution: Let and be three vectors such that is coplanar with and . If…

$\begin{aligned}& \overrightarrow{\mathrm{c}}=\lambda(\overrightarrow{\mathrm{b}} \times(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})) \\& =\lambda((\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{~b}}) \overrightarrow{\mathrm{a}}-(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{~b}}) \overrightarrow{\mathrm{b}}) \\& =\lambda(11 \overrightarrow{\mathrm{a}}-2 \overrightarrow{\mathrm{~b}})=\lambda(11 \mathrm{i}+22 \mathrm{j}+33 \mathrm{k}-6 \mathrm{i}-2 \mathrm{j}+2 \mathrm{k}) \\& =\lambda(5 \mathrm{i}+20 \mathrm{j}+35 \mathrm{k}) \\& =5 \lambda(5 \mathrm{i}+4 \mathrm{j}+7 \mathrm{k}) \\& =\text { Given } \overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}=5 \\& =5 \lambda(1+8+21)=5=\lambda=\frac{1}{30} \\& \Rightarrow \overrightarrow{\mathrm{c}}=\frac{1}{6}(\mathrm{i}+4 \mathrm{j}+7 \mathrm{k}) \\& |\overrightarrow{\mathrm{c}}|=\frac{\sqrt{1+16+49}}{6}=\sqrt{\frac{11}{6}}\end{aligned}$