prove that the angle between any two diagonals of a cube is cos-1(1/3) - Maths - Three Dimensional Geometry

Question

prove that the angle between any two diagonals of a cube is
cos-1(1/3)

Ajanta Trivedi · Accepted Answer

<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/content_ck_images/images/pic27_02_1(1)%20jpg"
style="width: 263px; height: 211px;">
The coordinates of O and E are
(0,0,0) and (a,a,a) respectively
therefore the direction cosines of OE are
      a-0a2+a2+a2,a-0a2+a2+a2,a-0a2+a2+a2 = a3a,a3a,a3a= 13,13,13
the coordinates of A and D are
(a,0,0) and (0,a,a) respectively
therefore the direction cosines of AD are
   0-aa2+a2+a2,a-0a2+a2+a2,a-0a2+a2+a2=-a3 a,a3 a,a3
a= -13,13,13

If theta is the angle between
diagonals OE and AD ;
therefore
cosθ=l1 l2+m1 m2+n1 n2=13 -13+13 13+13
13=-13+13+13=13⇒θ=cos-113

hope this helps you