Question

The angle between two diagonals of a cube is

• A. $30^\circ$
• B. $45^\circ$
• C. $\cos^{-1} \left(\frac{1}{3} \right)$
• D. $\cos^{-1} \left( \frac{1}{\sqrt{3}} \right)$

Hint:

The angle between any pair of diagonals can be shown to be Cosθ=1/3.
 

Answer 1

The Correct Option is C

The correct answer is Option C) \(\cos^{-1} \left(\frac{1}{3} \right)\)

O(0, 0, 0), A(a, 0, 0), B(0, a, 0), R(0, 0, a), D(a, a, 0), K(a, 0, a), L(0, a, a,), P(a, a, a) direction-cosines of diagonal Op is proportional to (a, a, a)

so, Op's direct cosines are: 

 

The angle between two diagonals of a cube is cos^–11/3.

Discover More Topics From This Chapter: Three-Dimensional Geometry

Answer 2

The correct answer is Option C) \(\cos^{-1} \left(\frac{1}{3} \right)\)

Real Life Applications

• The angle between the diagonals was employed in the design of the Millennium Bridge in London and Burj Khalifa to guarantee that the structure would be sturdy and wouldn't swing excessively in the wind. 
• It is also used in the design of some types of trusses to guarantee that the truss is sturdy and won't collapse under load.

Question can also be asked as

1. What is the angle between any two diagonals of a cube? 
2. What is the smallest angle between any two diagonals of a cube? 
3. How to find the angle between two diagonals of a cube? 
4. What is the formula for the angle between two diagonals of a cube? 
5. How to prove that the angle between two diagonals of a cube is 90 degrees?
6. What is the acute angle between two diagonals of a cube? 
7. What is the dihedral angle between two diagonals of a cube? 
8. What is the angle between the two main diagonals of a cube? 
9. What is the angle between the two space diagonals of a cube?

Answer 3

The correct answer is Option C) \(\cos^{-1} \left(\frac{1}{3} \right)\)

DC’s of diagonals OP and AQ are
1/√3,1/√3,1/√3and−1/√3,1/√3,1/√3

So, cos θ=(1/√3)(−1/√3)+(1/√3)(1/√3)+(1/√3)(1/√3)
=1/3

so, θ = cos^–1 (1/3).

Diagonal of a Cube Formula

The Diagonal of a Cube Formula can be denoted by:

The diagonal of a Cube = \(\begin{array}{l}\sqrt{3}x\end{array}\)
The primary diagonal of a Cube cuts through the center of the Cube; the diagonal of the Cube's face is not the main diagonal. The main diagonal of a cube can be determined by the help of multiplying the length of one side with the square root of 3 (it is also called the body diagonal of a cube).

Also Read: 

Related Topics
Addition of Vectors	Equation of a Plane in Three-Dimensional Space	Coplanar Vector