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

Amber

Written by: Amber

Reviewed by: Dan Finlay

Updated on

DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HL

Angle between two planes

How do I find the angle between two planes?

  • There are two pairs of equal angles formed when two planes intersect

  • The angles between two planes are equal to the angles between their normal vectors

  • The formula for an angle between two planes is cos theta equals fraction numerator bold italic n subscript 1 times bold italic n subscript 2 over denominator open vertical bar bold italic n subscript 1 close vertical bar open vertical bar bold italic n subscript 2 close vertical bar end fraction

    • bold italic n subscript 1 is a normal vector to one plane

    • bold italic n subscript 2 is a normal vector to the other plane

  • You can subtract one angle from 180° to find another angle

Diagram showing two intersecting planes with angle θ between them and their normal vectors. Labels explain the angle between planes and normal vectors.
Example of angles between two planes

Examiner Tips and Tricks

Always draw a sketch of the planes and their normals. This will make sure you do not confuse this method with the method for finding an angle between a line and a plane.

Worked Example

Find the acute angle between the two planes which can be defined by equations begin mathsize 16px style capital pi subscript 1 colon blank 2 x minus y plus 3 z equals blank 7 end style and begin mathsize 16px style capital pi subscript 2 colon blank x plus 2 y minus z equals 20 end style.

3-11-3-ib-hl-aa-angle-between-two-planes-we-solution-2

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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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