Angle Between Cube Diagonal and Edge

00:00 3 marks

  • Build Confidence
  • SL & HL
  • Tier 3
  • Trigonometry
  • 3D geometry
  • Spatial reasoning

In a cube with side length \(s\), the space diagonal connects two opposite vertices. Using vector methods, calculate the angle between this space diagonal and one of the cube’s edges. Express your answer in terms of inverse cosine.

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Hint

Recall that the dot product of two vectors \(\vec{u}\) and \(\vec{v}\) is given by \(\vec{u} \cdot \vec{v} = |\vec{u}|\,|\vec{v}|\cos(\theta)\).

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Complete Answer

\(\theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right)\)

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