3D Geometry
Distance from vertex of cube to space diagonal
MMTS_Full_Test_10
Grade 12
Question:
The perpendicular distance of a corner of a unit cube from a diagonal not passing through it is
(A) $\sqrt{\dfrac{5}{3}}$
(B) $\sqrt{\dfrac{1}{3}}$
(C) $\sqrt{\dfrac{2}{3}}$
(D) $\sqrt{2}$
Step-by-Step Solution
Key Concept: Place unit cube with one vertex at origin. Body diagonal through $(0,0,0)$: direction $\hat{i}+\hat{j}+\hat{k}$. For the corner $C=(0,0,1)$ not on this diagonal, find perpendicular distance.
$|CM|=\sqrt{|OC|^2-|OM|^2}=\sqrt{1-1/3}=\sqrt{2/3}$.
Correct Answer: (C) $\sqrt{\dfrac{2}{3}}$