Did this video help you?
Angles Between Lines & Planes (DP IB Maths: AA HL)
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
- A direction vector for the line (b)
- 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
- Use the formula
- Subtract this angle from 90° to find the acute angle between the line and the plane
- Subtract the angle from if working in radians
Examiner Tip
- Remember that if the scalar product is negative your answer will result in an obtuse angle
- Taking the absolute value of the scalar product will ensure that you get the acute angle as your answer
Worked example
Find the angle in radians between the line L with vector equation and the plane with Cartesian equation .
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°
Examiner Tip
- In your exam read the question carefully to see if you need to find the acute or obtuse angle
- When revising, get into the practice of double checking at the end of a question whether your angle is acute or obtuse and whether this fits the question
Worked example
Find the acute angle between the two planes which can be defined by equations and .
You've read 0 of your 10 free revision notes
Unlock more, it's free!
Did this page help you?