Question

Find the angle between a diagonal of a cube and one of its edges.

Answer 1

54.74°

Explanation:

Draw a well labelled cube to have a better representation of the question.

A cube is 3-dimensional, therefore it will have coordinates in x,y and z axis. The diagonal of a cube is from one corner to another corner.

Assuming that cube is unit cube, we need to find n angle at one of its edges will make with the diagonal.

I choose the the diagonal and one edge with the unit vector (1,0,0) and (1,1,1).

a = (1,1,1)

b = (1,0,0)

To calculate the angle between two vectors,

a.b = |a||b|cosθ

For simplicity, calculate the dot product, and the magnitudes. Then we will substitute the values of the dot product, and the magnitudes of the vectors to solve for the angle.

Calculating the dot product

a.b = (1,1,1) . (1,0,0)

= (1 × 1) + (1 × 0) (1 × 0)

= 1

Calculating the magnitudes

1. Magnitude of (1,1,1)

`|a| = \sqrt{1^(2) + 1^(2) + 1^(2)}`

`|a| = √(3)`

2. Magnitude of (1,0,0)

`|b| = \sqrt{1^(2) + 0^(2) + 0^(2)}`

`|b| = 1`

Calculating the angle between the two vectors

cosθ
`= (a.b)/(|a||b|)`

cosθ
`= (1)/(√(3)* 1)`

cosθ
`= (1)/(√(3))`

θ
`= cos^(_-1) (1)/(√(3))`

θ = 54.7356°

θ ≈ 54.74°