QJEEM24Mac21d: JEE Main 2024 Physics - Class 11 JEEM24Mac21d Solution: Let a unit vector which makes an angle of with and an angle of…

$\begin{aligned}& \text { Let } \vec{C}=a \hat{i}+b \hat{j}+c \hat{k} \\& (a \hat{i}+b \hat{j}+c \hat{k}) \cdot(2 \hat{i}+2 \hat{j}-\hat{k})=1 \times 3 \times \frac{1}{2} \\& 2 a+2 b-c=\frac{3}{2} \qquad \text{... (1)}\\& (a \hat{i}+b \hat{j}+c \hat{k}) \cdot(\hat{i}-\hat{k})=1 \times \sqrt{2} \times \frac{1}{\sqrt{2}} \\& a-c=1 \quad \text{... (2)}\\& a^2+b^2+c^2=1 \quad \text{... (3)}\\\end{aligned}$

Solving (1), (2) and (3)

$\begin{aligned}& a+2 b=\frac{1}{2} \\& a^2+b^2+(a-1)^2=1 \\& 2 a^2-2 a+b^2=0 \\& 2 a^2-2 a+\left(\frac{2 a-1}{4}\right)^2=0 \\& 32 a^2-32 a+4 a^2-4 a+1=0 \\& 36 a^2-36 a+1=0 \\& a=\frac{36 \pm \sqrt{(36)^2-4(36)}}{2 \times 36} \\& =\frac{1}{2} \pm \frac{\sqrt{2}}{3} \\& b=\frac{1-2 a}{4} \Rightarrow b=\frac{1 \pm \frac{2 \sqrt{2}}{3}-1}{4} \\& =\mp \frac{1}{3 \sqrt{2}} \\& C=-\frac{1}{2} \pm \frac{\sqrt{2}}{3}\end{aligned}$

$\begin{aligned}& C+\left(\frac{-1}{2} \hat{i}+\frac{1}{3 \sqrt{2}} \hat{j}-\frac{\sqrt{2}}{3} \hat{k}\right) \\& =\frac{\sqrt{2}}{3} \hat{i}-\frac{1}{2} \hat{k}\end{aligned}$