The angle between a diagonal of a cube and an adjacent edge is 54.75°;
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 54.75°.
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