The angle between two planes

The angle between two planes

Angle Between Two Planes Overview

  • The angle between two planes refers to the angle formed by their normal vectors.
  • Normal vectors are perpendicular (or ‘normal’) to their respective planes. It is this perpendicularity that provides the key to calculating the angle between the planes.
  • The angle between two planes can vary from 0° (when the planes are identical) to 180° (when the planes are opposite).

Determining Normal Vectors

  • Each plane in a three-dimensional space has an equation of the form Ax + By + Cz = D.
  • The coefficients A, B, and C from this equation represent the components of the normal vector.
  • For instance, for the plane 2x + 3y - z = 5, the normal vector is (2, 3, -1).

Calculating the Angle

  • The dot product formula for vectors, known as the scalar product, can be adapted to calculate the angle between two planes.

  • Given two normal vectors a and b, the cosine of the angle (θ) between them is given by:

    cos(θ) = (a . b) / (||a|| ||b||)

  • Here, ‘a . b’ represents the dot product of a and b, while **’   a   ‘** and **’   b   ‘** represent the magnitudes (lengths) of these vectors.

Angles Greater than 90°

  • The scalar product formula only gives the acute angle (less than or equal to 90°) between two planes.
  • To find the obtuse angle (greater than 90°), subtract the acute angle from 180°.

Example

  • Suppose we have two planes with equations 2x + y - z = 3 and x -2y + 3z = 4.
  • First, extract the normal vectors: (2, 1, -1) and (1, -2, 3).
  • Second, calculate the dot product and resultant magnitude, and apply the scalar product formula to find the acute angle.
  • Lastly, to find the obtuse angle, subtract the acute value from 180°.

Further Tips

  • Remember to always express your final answer in degrees unless the question specifically asks for a different unit.
  • Get comfortable with vector calculations, particularly calculating the dot product and the magnitude of a vector, as this forms the bulk of finding the angle between two planes.
  • Remember the process: find the normal vectors, perform the scalar product calculation, then adjust if an obtuse angle is required.
  • Also keep in mind the range of possible angles between two planes - 0° to 180° inclusive - to verify whether your final answer is sensible.