If theta is the angle, in degrees, between the longest diagonal of the cube and any one of the edges of the cube, then, cos(theta) = ? A. 1/2 B. 1/sqrt(3) C. 1/sqrt(2) D. sqrt(3)/2

GATE 2021 · General Aptitude · Cube · medium

Answer: cos(theta) = 1/sqrt(3)

1. Set up coordinate system: Place unit cube with one vertex at origin. Space diagonal goes from (0,0,0) to (1,1,1).
2. Write vectors for diagonal and edge: Space diagonal vector d = (1,1,1), |d| = sqrt(1+1+1) = sqrt(3). Edge vector e = (1,0,0), |e| = 1.
3. Apply dot product angle formula: d . e = 1*1 + 1*0 + 1*0 = 1. cos(theta) = 1 / (sqrt(3) * 1) = 1/sqrt(3).