To find the angle between a diagonal of a cube and one of its edges, we can use some basic geometry principles. Let’s start by visualizing a cube. The diagonal we are referring to runs from one corner of the cube to the opposite corner, while one of the cube’s edges connects two adjacent corners.
Assuming that the length of each edge of the cube is s, we can find the coordinates for the relevant points:
- Point A (0, 0, 0) – this is one corner of the cube,
- Point B (s, 0, 0) – this is next to Point A, along the edge,
- Point C (s, s, s) – this is the opposite corner of the cube, where the diagonal ends.
The edge (from Point A to Point B, for example) can be represented as a vector: AB = (s, 0, 0).
The diagonal (from Point A to Point C) can be represented as another vector: AC = (s, s, s).
Next, we will calculate the angle θ between these two vectors using the dot product formula, which is given by:
cos(θ) = (A · B) / (|A| |B|)
First, we need to find the dot product of vectors AB and AC:
A · B = (s, 0, 0) · (s, s, s) = s² + 0 + 0 = s².
Next, we find the magnitudes of the vectors:
- |AB| = √(s² + 0² + 0²) = s,
- |AC| = √(s² + s² + s²) = √(3s²) = s√3.
Now, we substitute back into the cosine formula:
cos(θ) = s² / (s * s√3) = 1/√3.
To find θ, take the arc cosine:
θ = cos⁻¹(1/√3).
This computes to approximately 54.74 degrees. When rounded to the nearest degree, the angle is:
55 degrees.
So, the angle between a diagonal of a cube and one of its edges is approximately 55 degrees.