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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector PlanesEquation of a Plane in Cartesian Form

Equation of a Plane in Cartesian Form (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

Equation of a plane in cartesian form

How do I find the vector equation of a plane using a normal vector?

  • The formula for finding the vector equation of a plane is bold italic r times bold italic n equals bold italic r times bold italic a

    • bold italic r is the position vector of any point on the plane

    • bold italic a is the position vector of a known point on the plane

    • bold italic n is a normal vector which is perpendicular to the plane

Examiner Tips and Tricks

This is given in the formula booklet under the geometry and trigonometry section. However, you need to remember what the components represent.

  • A normal vector can be found by taking the vector product of two non-parallel vectors which are parallel to the plane

    • For bold italic r equals bold italic a plus lambda bold italic b plus mu bold italic c use bold italic n equals bold italic b cross times bold italic c

How do I find the vector equation of a plane in Cartesian form?

  • The formula for finding the Cartesian equation of a plane is a x plus b y plus c z equals d

    • The vector open parentheses table row a row b row c end table close parentheses is perpendicular to the plane

Examiner Tips and Tricks

This is given in the formula booklet under the geometry and trigonometry section. However, you need to remember how to find the values of the coefficients.

  • Cartesian equations represent the same planes if they are scalar multiples of each other

    • For example, 2 x plus 3 y plus z equals 5 and 4 x plus 6 y plus 2 z equals 10 represent the same plane

    • For example, 2 x plus 3 y plus z equals 5 and 4 x plus 6 y plus 2 z equals 9 represent different planes

How do I find the equation of a plane in Cartesian form given the vector form?

  • Suppose you are given the equation bold italic r equals bold italic a plus lambda bold italic b plus mu bold italic c

  • STEP 1
    Use the vector product to find a normal vector to the plane

    • bold italic n equals bold italic b cross times bold italic c

  • STEP 2
    Write the expression a x plus b y plus c z

    • open parentheses table row a row b row c end table close parentheses equals bold italic n

  • STEP 3
    Use the scalar product of the normal vector and the position vector to find the value of d

    • d equals bold italic n times bold italic a

Examiner Tips and Tricks

Step 3 is equivalent to substituting the coordinates of a point on the plane into the expression a x plus b y plus c z.

How do I determine whether a point lies on a plane?

  • You can do this easily using the Cartesian equation

  • Substitute the coordinates of the point into the Cartesian equation

    • If the equation is satisfied, then the point lies on the plane

    • Otherwise, it does not lie on the plane

Worked Example

A plane straight capital pi contains the point Aleft parenthesis 2 comma space 6 comma negative 3 right parenthesis and has a normal vector open parentheses table row 3 row cell negative 1 end cell row 4 end table close parentheses.

a) Find the equation of the plane in its Cartesian form.

JMMFvkWx_3-11-1-ib-aa-hl-vector-plane-cartesian-we-solution-a

b) Determine whether point B with coordinates left parenthesis negative 1 comma space 0 comma negative 2 right parenthesis lies on the same plane.

ghYsvxS~_3-11-1-ib-aa-hl-vector-plane-cartesian-we-solution-b

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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.