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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector PlanesAngles Between Two Planes

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

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

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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

    • cos theta equals fraction numerator bold italic n subscript bold 1 times bold italic n subscript bold 2 over denominator open vertical bar bold italic n subscript bold 1 close vertical bar open vertical bar bold italic n subscript bold 2 close vertical bar end fraction

  • 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°

3-11-3-ib-hl-aa-angle-between-two-planes-diagram-2

Examiner Tips and Tricks

  • 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 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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    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.