JEE Main 2024 (Online) 29th January Evening ShiftQuestion 1
Let a unit vector $\hat u=x\hat i+y\hat j+z\hat k$ make angles $\frac\pi2,\frac\pi3$ and $\frac{2\pi}{3}$ with the vectors $\frac1{\sqrt2}\hat i+\frac1{\sqrt2}\hat k$, $\frac1{\sqrt2}\hat j+\frac1{\sqrt2}\hat k$ and $\frac1{\sqrt2}\hat i+\frac1{\sqrt2}\hat j$ respectively. If $\vec v=\frac1{\sqrt2}\hat i+\frac1{\sqrt2}\hat j+\frac1{\sqrt2}\hat k$, then $|\hat u-\vec v|^{2}$ is equal to
Vector Algebra
Solution
Conditions: $x+z=0$, $\frac{y+z}{\sqrt2}=\frac12$, $\frac{x+y}{\sqrt2}=-\frac12$, so $\hat u=(-\frac1{\sqrt2},0,\frac1{\sqrt2})$. $\hat u-\vec v=(-\sqrt2,-\frac1{\sqrt2},0)$, $|\hat u-\vec v|^2=2+\frac12=\frac52$.