Angles Between Lines & Planes (DP IB Analysis & Approaches (AA)): Revision Note

Angle Between Line & Plane

What is meant by the angle between a line and a plane?

• When you find the angle between a line and a plane you will be finding the angle between the line itself and the line on the plane that creates the smallest angle with it

  • This means the line on the plane directly under the line as it joins the plane

• It is easiest to think of these two lines making a right-triangle with the normal vector to the plane

  • The line joining the plane will be the hypotenuse

  • The line on the plane will be adjacent to the angle

  • The normal will the opposite the angle

How do I find the angle between a line and a plane?

• You need to know:

  • A direction vector for the line (b)

    • This can easily be identified if the equation of the line is in the form 

  • A normal vector to the plane (n)

    • This can easily be identified if the equation of the plane is in the form 

• Find the acute angle between the direction of the line and the normal to the plane

  • Use the formula 

    • The absolute value of the scalar product ensures that the angle is acute

• Subtract this angle from 90° to find the acute angle between the line and the plane

  • Subtract the angle from  if working in radians

Angle Between Two Planes

How do I find the angle between two planes?

• The angle between two planes is equal to the angle between their normal vectors

  • It can be found using the scalar product of their normal vectors

  • 

• If two planes Π₁ and Π₂ with normal vectors n₁ and n₂ meet at an angle then the two planes and the two normal vectors will form a quadrilateral

  • The angles between the planes and the normal will both be 90°

  • The angle between the two planes and the angle opposite it (between the two normal vectors) will add up to 180°