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3D Geometry

Distance from vertex of cube to space diagonal

MJMT_Full_Test_09

Grade 12

Question:

The perpendicular distance of a corner of a unit cube from a diagonal not passing through it is

$\sqrt{\dfrac{5}{3}}$

$\sqrt{\dfrac{1}{3}}$

$\sqrt{\dfrac{2}{3}}$

$\sqrt{2}$

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: 3