Answer to Puzzle #44: Diagonals on the Face of a Cube. What's the Angle?

cube with a line drawn across two faces to meet at one corner44. The two red lines in the diagram shown are drawn diagonally on the faces of a cube. What is the angle between them at the point where they join?

I got this puzzle from an article on BBC news called 'Martin Gardner, puzzle master extraordinaire.' More spacial reasoning than a puzzle in the conventional sense but let's take a look.

Before reading the answer can I interest you in a clue?

It might be that if you are in a particularly mathematical frame of mind you are trying to use a three dimensional co-ordinate system and a matrix transform to project the 45° angles in the corners in to a different plane. By all means. But it's not necessary, essentially this is a 2D problem, it's just about finding the right two dimensions to work in. Let's add another line to the diagram and see how we go.
cube with a line drawn across two faces to meet at one corner and a dotted line
Any clearer? The dotted line completes a triangle. What's more since all it's edges are made up of diagonals from the squares, they are all the same length, it is an equilateral triangle. Like all equilateral triangles it's internal angles are each 60°

A problem I have found with these 3D nets is not everyone sees them the same, it's like an optical illusion. So at some point over the next few images it should become clear what I'm driving at...
cube with a section removed slightly
cube with a section removed fully
cube with viewed in the plane of the triangle
cube with viewed in the plane of the triangle with section removed
And finally, just because we can. cube with viewed in the plane of the triangle with section removed and answer drawn on.

Vector Maths

As I've said before it's my intention that anyone using this site does not need formal maths training. But where there is an obvious maths solution I will try to include it.
cube showing axis xyz
The above diagram shows a standard right handed co-ordinate system with the label on the z axis hidden. We will calculate the angle of the two red lines that meet at the origin. The first being defined as (1,1,0) and the second (1,0,1)

The angle between to vectors is given by:
cos θ = (u•v) / (|u||v|)
cos θ = (1•1 + 1•0 + 0•1) / (√2√2)
cos θ = 1/2
⇒ θ = π/3 or 60°






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