Q. 39 - Calculus

Question

Find the angle between the diagonal of a face of a cube and the adjoining edge of the cube that is not an edge of that face.

Step-by-Step Solution

Answer

A

Step 1: Set up coordinates

Place the cube with vertices at the origin with side length 1. Consider the face diagonal on the bottom face from \((0,0,0)\) to \((1,1,0)\), and the adjacent edge from \((0,0,0)\) to \((0,0,1)\).

Step 2: Compute the angle

Face diagonal vector: \(\mathbf{d} = (1,1,0)\), \(|\mathbf{d}| = \sqrt{2}\).
Edge vector: \(\mathbf{e} = (0,0,1)\), \(|\mathbf{e}| = 1\).
\(\cos\theta = \frac{\mathbf{d} \cdot \mathbf{e}}{|\mathbf{d}||\mathbf{e}|} = \frac{0}{\sqrt{2}} = 0\)
\(\theta = 90°\).
However, if the edge is not perpendicular to the face containing the diagonal (e.g., edge \((1,0,0)\) and diagonal \((1,1,0)\)): \(\cos\theta = \frac{1}{\sqrt{2}}\), so \(\theta = 45°\).