Q 11-03-171JEE MainJEE Main 2018 (16 Apr, Shift 1)Medium
Let $\vec A = \hat i + \hat j$ and $\vec B = 2\hat i - \hat j$. The magnitude of a coplanar vector $\vec C$ such that $\vec A\cdot\vec C = \vec B\cdot\vec C = \vec A\cdot\vec B$ is given by:
Answer: (C) $\sqrt{\dfrac{5}{9}}$
$\vec A\cdot\vec B = 2 - 1 = 1$. Let $\vec C = x\hat i + y\hat j$ (coplanar with $\vec A$ and $\vec B$):
$$\vec A\cdot\vec C = x + y = 1, \qquad \vec B\cdot\vec C = 2x - y = 1$$
Adding: $3x = 2$, so $x = \dfrac23$ and $y = \dfrac13$.
$$|\vec C| = \sqrt{\frac49 + \frac19} = \sqrt{\frac59}$$
Solution by Sreeraj P, M.Sc Physics