The angle between two planes

The Angle Between Two Planes

Understanding the Concept

• Planes in mathematics are flat, two-dimensional surfaces that extend infinitely far in every direction.
• Every plane can be represented by a normal vector, which is a vector that is perpendicular to the plane.
• The angle between two planes is the angle between their normal vectors.

The Dot Product of Two Vectors

• The dot product of two vectors is given by the formula **A.B =

  A

  B

  cos θ**, where A and B are the vectors,

  A

  and

  B

  are their magnitudes and θ is the angle between them.

• We can rearrange this formula to find θ: **θ = acos(A.B / (

  A

  B

  ))**.

Finding the Angle Between Two Planes

• To find the angle between two planes, we need to find the angle between their normal vectors.
• Firstly, find the normal vectors for each plane.
• Secondly, calculate the dot product of these two vectors.
• Thirdly, calculate the magnitudes of both vectors.

• Finally, substitute these values into the formula **θ = acos(A.B / (

  A

  B

  ))** to find the angle between the planes.

Acute and Obtuse Angles Between Planes

• The angle between two planes can either be acute (less than 90 degrees) or obtuse (more than 90 degrees).

• The formula **θ = acos(A.B / (

  A

  B

  ))** will always give you the acute angle.

• To find the obtuse angle, subtract the acute angle from 180 degrees.

Real World Applications

• Understanding the angle between two planes is crucial in fields such as architecture, physics, and computer graphics.

Reviewing Work

• Be careful when calculating dot products and magnitudes, as small mistakes can lead to significant inaccuracies.
• Always remember that the angle is between the normal vectors of the planes, not the planes themselves.
• Practise finding the angle between two planes regularly in order to become familiar with the process.