Question

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

• A. $\dfrac{1}{2}$
• B. $\dfrac{1}{\sqrt{3}}$
• C. $\dfrac{1}{\sqrt{2}}$
• D. $\dfrac{\sqrt{3}}{2}$

Hint:

The space diagonal of a cube has direction ratios $(1,1,1)$, which makes vector methods very effective.

Answer 1

The Correct Option is B

Step 1: Represent the cube using vectors.
Consider a cube of side length $a$. Take one edge of the cube along the $x$-axis, represented by the vector \[ \vec{e} = (a, 0, 0). \] The longest diagonal of the cube connects opposite vertices and is represented by the vector \[ \vec{d} = (a, a, a). \]

Step 2: Use the dot product formula.
The cosine of the angle $\theta$ between two vectors $\vec{e}$ and $\vec{d}$ is given by \[ \cos \theta = \frac{\vec{e} \cdot \vec{d}}{|\vec{e}|\,|\vec{d}|}. \]

Step 3: Compute dot product and magnitudes.
\[ \vec{e} \cdot \vec{d} = a^2, |\vec{e}| = a, |\vec{d}| = a\sqrt{3}. \]

Step 4: Evaluate $\cos \theta$.
\[ \cos \theta = \frac{a^2}{a \cdot a\sqrt{3}} = \frac{1}{\sqrt{3}}. \]

Step 5: Conclusion.
Hence, the correct value of $\cos \theta$ is $\dfrac{1}{\sqrt{3}}$.