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

To solve the problem of finding the angle \(\theta\) between the longest diagonal of a cube and one of its edges, let's consider a cube with side length a.

The formula for the length of the diagonal across a cube is given by the space diagonal formula:

• d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3}

The length of any edge of the cube is simply a.

Now, to find \cos \theta, where \theta is the angle between the space diagonal and any cube edge, we use the cosine formula:

• \(\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}\)
• In this case,
  \(\cos \theta = \frac{a}{a\sqrt{3}} = \frac{1}{\sqrt{3}}\)

Given the options:

• \(\frac{1}{2}\)
• \(\frac{1}{\sqrt{3}}\)
• \(\frac{1}{\sqrt{2}}\)
• \(\frac{\sqrt{3}}{2}\)

The correct answer is \(\frac{1}{\sqrt{3}}\) because the calculation for \cos \theta results in this value. Thus, the correct answer is:

• \(\frac{1}{\sqrt{3}}\)

Conclusion: The angle \theta between the longest diagonal of the cube and any one of its edges has a cosine value of \frac{1}{\sqrt{3}}. This approach utilizes the relationship between the space diagonal and the cube's edge, supported by cosine rule.