Find the angle between a diagonal of a cube and one of its edges. (Round your answer to the nearest degree).

Solution:

We have to find the angle between a diagonal of a cube and one of its edges.

A cube is a 3D solid object with six square faces and all the sides of a cube are of the same length.

The diagonal of a cube is the line segment that connects any two non-adjacent vertices of the cube.

The edge of a cube is a line segment joining two vertices of the cube.

Assuming the length of a side of the cube as 1 unit, the three sides are given by the vectors

a = (1, 0, 0)

b = (0, 1, 0)

c = (0, 0, 1)

The diagonal is given by the vector, v = (1, 1, 1)

The angle between the diagonal and one edge of the cube is given by

cos θ = v.a / |v|.|a|

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

= 1(1) + 1(0) + 1(0)

= 1 + 0 + 0

v.a = 1

Magnitude of vector V = √[(1)² + (1)² + (1)²]

|v| = √3

Similarly, |a| = √1

So, |v|.|a| = √3.√1 = √3

Now, cos θ = 1/√3

So, θ = cos¹(1/√3)

θ = 54.75°

Therefore, the required angle is 55°

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Find the angle between a diagonal of a cube and one of its edges. (Round your answer to the nearest degree).

Summary:

The angle between a diagonal of a cube and one of its edges is 55°.