Angle Between a Line and a Plane (3D) Calculator

Published on: June 7, 2026
Final Answer: Free • Full Steps: Pro

This Angle Between a Line and a Plane (3D) Calculator helps you find the angle between a line and a plane in three-dimensional space. Use the line’s direction vector and the plane’s normal vector, then apply the angle formula to calculate the result. This shows how the line is inclined relative to the plane. It is a simple way to check answers, understand the method clearly, and practise 3D coordinate geometry step by step.

Step-by-step method

  1. Identify the line direction vector v and the plane normal vector n.
  2. Compute the dot product |v · n|.
  3. Compute the magnitudes |v| and |n|.
  4. Use sin(α) = |v · n| / (|v||n|) and solve for α.

Formula:

\(\sin\left(\alpha\right) = \frac{\left|\vec{v} \cdot \vec{n}\right|}{\left|\vec{v}\right|\left|\vec{n}\right|}\)