Angle Between Two Planes (3D)

Published on: June 14, 2026
Final Answer: Free • Full Steps: Pro

This calculator finds the angle between two planes in 3D. Enter input like Ax+By+Cz=d; A2x+B2y+C2z=d2. Example: 2x-3y+4z=10; x+4y-2z=7.

Step-by-step method

  1. Identify the normal vectors n₁ and n₂ from the two planes.
  2. Compute the dot product |n₁ · n₂|.
  3. Compute the magnitudes |n₁| and |n₂|.
  4. Use cos(θ) = |n₁ · n₂| / (|n₁||n₂|) and solve for θ.

Formula:

\(\cos\left(\theta\right) = \frac{\left|\vec{n}_{1} \cdot \vec{n}_{2}\right|}{\left|\vec{n}_{1}\right|\left|\vec{n}_{2}\right|}\)