If \(\theta\) is the angle, in degrees, between the longest diagonal of the…
2021
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.
\(\frac {1}{2}\) - B.
\(\frac {1}{\sqrt 3}\) - C.
\(\frac {1}{\sqrt 2}\) - D.
\(\frac {\sqrt 3}{2}\)
Attempted by 44 students.
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Correct answer: B
Key idea: represent the space diagonal and an edge as vectors and use the dot product to find the cosine of the angle between them.
Step 1: Take the cube with side length 1. The space (longest) diagonal is the vector (1,1,1); take an edge along the x-axis as the vector (1,0,0).
Step 2: Compute the dot product: (1,1,1)·(1,0,0) = 1.
Step 3: Compute magnitudes: |(1,1,1)| = √3, |(1,0,0)| = 1.
Step 4: Use the dot product formula cosθ = (u·v)/(|u||v|) = 1/(√3·1) = 1/√3.
Therefore cosθ = 1/√3.