GATE CS 2021 Set 2 Q3 — General Aptitude

To find the angle $\theta$ between the longest diagonal of a cube and any of its edges, we can place the cube in a 3D Cartesian coordinate system.

Let one vertex of the cube of side length $a$ be at the origin $(0, 0, 0)$.

• The edge along the x-axis can be represented by the vector:
  $\vec{u} = (a, 0, 0)$
• The longest diagonal from $(0,0,0)$ to $(a, a, a)$ is represented by the vector:
  $\vec{d} = (a, a, a)$

The cosine of the angle $\theta$ between these two vectors is given by the dot product formula:

Computing the individual terms:

• $\vec{d} \cdot \vec{u} = (a)(a) + (a)(0) + (a)(0) = a^2$
• $|\vec{d}| = \sqrt{a^2 + a^2 + a^2} = a\sqrt{3}$
• $|\vec{u}| = \sqrt{a^2 + 0^2 + 0^2} = a$

Substituting these values back into the formula:

Thus, the correct option is B.