The diagonals of a cube meet at the center of the cube. To determine the angle between two diagonals, consider the geometric properties of the cube.
A cube has 12 face diagonals and 4 space diagonals. We are interested in the space diagonals because they span from one vertex of the cube to the opposite vertex.
Let's place the cube in a 3-dimensional Cartesian coordinate system with vertices at:
(0,0,0),(1,0,0),(1,1,0),(0,1,0),(0,0,1),(1,0,1),(1,1,1), and (0,1,1).
Consider the space diagonals of the cube, such as the one joining
(0,0,0) to
(1,1,1) and another joining
(1,0,0) to
(0,1,1). The direction vectors of these diagonals can be written as:
‌1=⟨1,1,1⟩‌2=⟨−1,1,1⟩ We use the dot product formula for vectors to find the angle between them,
θ, which is given by:
1⋅2=|1||2|‌cos‌θ The dot product of these vectors is:
The magnitudes of the vectors are:
‌|1|=√12+12+12=√3‌|2|=√(−1)2+12+12=√3 Substitute these values into the dot product formula:
cos‌θ=‌=‌=‌Therefore, the angle
θ is:
θ=cos−1(‌)Hence, the correct option is:
Option A:
cos−1(‌)