JEE Main · 2024 · 31 Jan 2024 · Shift 2 · Three Dimensional Geometry

$\overrightarrow{AB}=B-A=(12,4,6)$, which simplifies to direction $(6,2,3)$ with magnitude $\sqrt{36+4+9}=\sqrt{49}=7$.

The unit vector along $AB$ is $\frac{(6,2,3)}{7}$.

Moving a distance $21$ from $A$ along this direction:

$P=A+21\cdot \frac{(6,2,3)}{7}=A+3(6,2,3)=(4+18,-6+6,-2+9)=(22,0,7)$.

(Moving in the opposite direction would give negative coordinates, which is not allowed since $a,b,c$ must be non-negative, so this is the only valid point.)

Distance from $P(22,0,7)$ to $Q(4,-12,3)$:

$PQ=\sqrt{(22-4{)}^2+(0+12{)}^2+(7-3{)}^2}=\sqrt{324+144+16}=\sqrt{484}=22$.