# Why do the face diagonals have different angles?

In summary, the conversation discusses finding the angle of face diagonals in a unit cube using the dot product and cos-1 formula. The attempted solution is incorrect because it uses an edge and an inside diagonal instead of two face diagonals. The correct method is to find the angle between two face diagonals originating from the same base-point and lying in two of the faces that meet at that point.

## Homework Statement

Find the angle of the face diagonals of a (unit) cube.

I agree with this solution, but I have a problem with another face diagonal: the face diagonal from the angle (0,0,1),(0,0,0), and (1,1,1).

## Homework Equations

dot product
cos-1(a.b/ (|a||b|)

## The Attempt at a Solution

From the solution, we have an angle given from the points (1,0,1), (0,0,0), and (0,1,1).
Using the def. of dot product, if A is the vector (0,0,0) to (1,0,1) and B is the vector (0,0,0) to (0,1,1)
cos-1(A.B/(|A||B|)), where A.B = 1, |A| = |B| = sqrt(2).
Thus cos-1(1/ [ (sqrt(2)sqrt(2) ]) = 60 deg. Ok.

the face diagonal from the angle (0,0,1),(0,0,0), and (1,1,1):
A is the vector (0,0,0) to (0,0,1) and B is the vector (0,0,0) to (1,1,1)
cos-1(A.B/(|A||B|)), where A.B = 1, |A| = 1, |B| = sqrt(3).
thus cos-1(1/ [ sqrt(3) ]) != 60 deg.

Why is this happening?

The line from (0,0,0) to (0,0,1) is an edge, not a face diagonal. The line from (0,0,0) to (1,1,1) is an "inside" diagonal, not a face diagonal.

Isn't the line from (0,0,1) to (1,1,1) a face diagonal?
Fredrik said:
The line from (0,0,0) to (0,0,1) is an edge, not a face diagonal. The line from (0,0,0) to (1,1,1) is an "inside" diagonal, not a face diagonal.
The angle made with the edge and the inside diagonal is the angle of the face diagonal, assuming the face diagonal is (0,0,1) to (1,1,1), right?

Isn't the line from (0,0,1) to (1,1,1) a face diagonal?

The angle made with the edge and the inside diagonal is the angle of the face diagonal, assuming the face diagonal is (0,0,1) to (1,1,1), right?

Wrong: look again at the diagram. You want the angle between *two* face-diagonals originating at the same base-point and lying in two of the faces that meet at that point. So, the vector from (0,0,1) to (1,1,1) is a face diagonal. but the vector from (0,0,1) to (0,0,0) is not.

Ray Vickson said:
You want the angle between *two* face-diagonals originating at the same base-point and lying in two of the faces that meet at that point.
Oh so that's what face diagonals mean. Thank you.

Isn't the line from (0,0,1) to (1,1,1) a face diagonal?
Yes it is. But you're looking for the angle between the two lines I mentioned, and they are not face diagonals.

## 1. Why do the face diagonals have different angles?

The angles of the face diagonals are determined by the shape and dimensions of the face. In a rectangular face, the diagonals will always have different angles because the shape is not symmetrical.

## 2. How does the shape of the face affect the angles of the diagonals?

The shape of the face directly affects the angles of the diagonals. In a square face, for example, all four angles of the diagonals will be equal because the shape is symmetrical. However, in a rectangular face, the angles will differ because the shape is not symmetrical.

## 3. Is there a mathematical explanation for why the face diagonals have different angles?

Yes, there is a mathematical explanation for why the face diagonals have different angles. It involves using the Pythagorean theorem to calculate the length of the diagonals and then using trigonometric functions to determine the angles.

## 4. Are there any real-world applications for understanding the angles of face diagonals?

Understanding the angles of face diagonals is important in fields such as architecture and engineering. It helps in designing and constructing structures with precise and stable dimensions, which is crucial for their structural integrity.

## 5. Can the angles of face diagonals be the same in any face shape?

Yes, in certain face shapes such as a square or a regular polygon, the angles of the face diagonals can be the same. This is because these shapes have equal sides and angles, making them symmetrical and resulting in equal diagonal angles.

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