Question

Coordinate planes and the planes \( \pi_1, \pi_2, \pi_3 \), which are respectively parallel to YZ, ZX, XY planes at distances a, b, c, form a rectangular parallelepiped. If \( d_1 \) is a diagonal of the face on XY-plane not passing through origin and \( d_2 \) is diagonal of plane \( \pi_2 \), then the cosine of the angle between \( d_1 \) and \( d_2 \) is:

• A. \( \sqrt{a^2 + b^2} \)
• B. \( \frac{a}{\sqrt{a^2 + b^2 + c^2}} \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{\sqrt{a^2 + b^2}}{\sqrt{a^2 + b^2 + c^2}} \)

Hint:

To find the angle between diagonals in a parallelepiped, use the dot product formula and consider the geometric relations between the vectors.

Answer 1

The Correct Option is D

The cosine of the angle between two vectors is given by the dot product formula. By considering the geometry of the rectangular parallelepiped and the diagonals, we get the cosine as \( \frac{\sqrt{a^2 + b^2}}{\sqrt{a^2 + b^2 + c^2}} \).