By Expressing diagonals as vectors, and using the definition of the dot product, find the smaller angle between any two diagonals of a cube, where each diagonal connects diametrically opposite corners and passes through the center of the cube. Solution vectors of diagonal will be from (0,0,0) to (a,a,a) d1 = ai +aj + ak from( a,0,0) to (0,a,a) d2 = -ai + aj + ak cos = d1 . d2 / { |d1| |d2| } d1.d2 = -a^2 + a^2 + a^2 =a^2 |d1| = 3 a |d2| = 3 a cos = a^2 /3 a^2 = 1/3 = cos-1(1/3).

1. By Expressing diagonals as vectors, and using the definition of the dot product, find the smaller
angle between any two diagonals of a cube, where each diagonal connects diametrically opposite
corners and passes through the center of the cube.
Solution
vectors of diagonal will be
from (0,0,0) to (a,a,a)
d1 = ai +aj + ak
from( a,0,0) to (0,a,a)
d2 = -ai + aj + ak
cos = d1 . d2 / { |d1| |d2| }
d1.d2 = -a^2 + a^2 + a^2 =a^2
|d1| = 3 a
|d2| = 3 a
cos = a^2 /3 a^2 = 1/3
= cos-1(1/3)