GATE CS 2021 Set 2 Q.3 (GA Q.3): Quantitative Aptitude

If $\theta$ is the angle, in degrees, between the longest diagonal of the cube and any one of the edges of the cube, then, $\cos \theta =$

$\frac{1}{2}$

$\frac{1}{\sqrt{3}}$

$\frac{1}{\sqrt{2}}$

$\frac{\sqrt{3}}{2}$

Official answer and solution

Official answer: B
From: Official GATE 2021 answer key

Solution

Use 3D coordinate geometry to determine the cosine of the angle between a body diagonal and an edge of a cube.

Place a cube of edge length $a$ in a Cartesian coordinate system with one vertex at the origin $(0, 0, 0)$ . Let the edges along the axes be represented by vector $\vec{u} = a\hat{i} = (a, 0, 0)$ .

The longest diagonal (body diagonal) connecting $(0, 0, 0)$ to $(a, a, a)$ is given by vector $\vec{v} = a\hat{i} + a\hat{j} + a\hat{k}$ .

The angle $\theta$ between $\vec{u}$ and $\vec{v}$ satisfies:

\cos \theta = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \|\vec{v}\|} = \frac{a \cdot a + 0 + 0}{\sqrt{a^2} \cdot \sqrt{a^2 + a^2 + a^2}} = \frac{a^2}{a \cdot a\sqrt{3}} = \frac{1}{\sqrt{3}}

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2021 Computer Science and Information Technology - Session 2 (CS2)
Question: 3 of 65, printed as GA Q.3
Section: General Aptitude

Report an error in this question

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.

Question

Solution

Source