Angle between diagonals of a cube

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In summary, the problem is to find the acute angle between two diagonals of a cube. The length of a diagonal of a cube with side lengths of one is sqrt(3). One possible approach is to use unit vectors and find the angle between them, but this may not give the correct answer. Another approach is to locate the cube on a coordinate axis, determine the coordinates of its corners, and use the definition of the dot product to find the angle. Drawing a cube and sketching in the diagonals can also help visualize the problem.
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BoundByAxioms
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Homework Statement


Find the acute angle between two diagonals of a cube.


Homework Equations


N/A


The Attempt at a Solution


I know that the length of a diagonal of a cube whose side lengths are each one is sqrt(3), so I think it has something to do with that. Other than that, I'm drawing a blank. I could use the unit vectors <1,0,0> and <0,1,0> and find the angle between them, but that's not giving me the right answer.
 
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  • #2
Here's something you can try. Locate the cube on a coordinate axis, and determine the coordinates of is corners. Then find the vectors corresponding to the cube diagonals and make use of the definition of the dot product.
 
  • #3
Draw a cube and sketch in a face diagonal and a space diagonal.
Then draw two triangles and smile.
 

FAQ: Angle between diagonals of a cube

1. What is the angle between diagonals of a cube?

The angle between diagonals of a cube is equal to 109.5 degrees. This is the same for any type of cube, regardless of its size or orientation.

2. How is the angle between diagonals of a cube calculated?

The angle between diagonals of a cube can be calculated using the formula arccos(1/3) or approximately 109.5 degrees. This formula is derived from the cosine law, which is used to calculate angles in triangles.

3. Is the angle between diagonals of a cube the same as the angle between its edges?

No, the angle between diagonals of a cube is not the same as the angle between its edges. The angle between edges of a cube is 90 degrees, while the angle between diagonals is approximately 109.5 degrees.

4. Why is the angle between diagonals of a cube important?

The angle between diagonals of a cube is important in geometry and mathematics because it is used to calculate the diagonal length of a cube. It is also useful in understanding the symmetry and properties of a cube.

5. Does the angle between diagonals of a cube change if the cube is skewed or distorted?

No, the angle between diagonals of a cube remains constant at 109.5 degrees, even if the cube is skewed or distorted. This is because the formula used to calculate the angle is based on the size and shape of the cube, not its orientation.

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