JEE Main 2024 (Online) 29th January Morning ShiftQuestion 1
Let $O$ be the origin and the position vectors of $A$ and $B$ be $2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$ respectively. If the internal bisector of $\angle AOB$ meets the line $AB$ at $C$, then the length of $OC$ is
Vector Algebra
Solution
$|OA|=3$, $|OB|=\sqrt{4+16+16}=6$. The bisector divides $AB$ in ratio $AC:CB=3:6=1:2$. $\vec{OC}=\dfrac{2\vec{OA}+\vec{OB}}{3}=\dfrac{(6,8,6)}{3}=(2,\tfrac83,2)$. $|OC|^2=4+\tfrac{64}{9}+4=\tfrac{136}{9}\Rightarrow|OC|=\tfrac23\sqrt{34}$. Option (A).