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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector Equations of LinesEquation of a Line in Vector Form

Equation of a Line in Vector Form (DP IB Analysis & Approaches (AA)): Revision Note

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Reviewed by: Dan Finlay

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DP Analysis & Approaches (AA): HLDP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HLDP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HL

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Equation of a Line in Vector Form

How do I find the vector equation of a line?

  • The formula for finding the vector equation of a line is bold italic r equals bold italic a plus lambda bold italic b

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

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

    • bold italic b is a direction (displacement) vector

    • lambda is a scalar

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.

You can compare it to the Cartesian equation of a 2D line, y equals m x plus c. The component by itself is a point on the line and the component multiplied by the variable is the direction.

  • There are an infinite number of ways to write the equation

    • There are an infinite number of options for bold italic a

    • Any scalar multiple of a direction vector is also a direction vector

How do I find the vector equation of a line that passes through two points?

  • Suppose a line passes through the points with position vectors bold italic a and bold italic p

  • Find a direction vector

    • Both bold italic a minus bold italic p and bold italic p minus bold italic a are direction vectors

  • Use the given formula

    • bold italic r equals bold italic a plus-or-minus lambda open parentheses bold italic a minus bold italic p close parentheses

    • bold italic r equals bold italic p plus-or-minus lambda open parentheses bold italic a minus bold italic p close parentheses

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

  • To check if the position vector bold italic p lies on the line bold italic r equals bold italic a plus lambda bold italic b

    • Substitute bold italic r equals bold italic p into the equation

      • bold italic p equals bold italic a plus lambda bold italic b

    • Check to see if there is a value of lambda which makes the equation true

    • This is the same as checking if bold italic p minus bold italic a is a scalar multiple of bold italic b

      • bold italic p minus bold italic a equals lambda bold italic b

  • You can also use algebra

    • Write the components of the vectors in the equation

      • bold italic r blank equals blank open parentheses fraction numerator a subscript 1 over denominator table row cell a subscript 2 end cell row cell a subscript 3 end cell end table end fraction close parentheses plus lambda open parentheses fraction numerator b subscript 1 over denominator table row cell b subscript 2 end cell row cell b subscript 3 end cell end table end fraction close parentheses

    • Write the components of the position vector of the point to test

      • bold italic p equals blank open parentheses fraction numerator p subscript 1 over denominator table row cell p subscript 2 end cell row cell p subscript 3 end cell end table end fraction close parentheses 

    • Form a system of linear equations

      • p subscript 1 equals blank a subscript 1 plus lambda b subscript 1

      • p subscript 2 equals blank a subscript 2 plus lambda b subscript 2

      • p subscript 3 equals blank a subscript 3 plus lambda b subscript 3

    • Solve one of the equations to find a value of lambda

    • Check that this value also satisfies the other two equations

      • If it does, then the point lies on the line

      • Otherwise, the point does not lie on the line

Examiner Tips and Tricks

There are an infinite number of ways to write the equation. Therefore, your answer might look different to the mark scheme's answer. However, you could still be correct.

Worked Example

a) Find a vector equation of a straight line through the points with position vectors a = 4i – 5k and b = 3i - 3k

M83a0TRO_3-10-1-ib-aa-hl-vector-equation-of-a-line-we-a

b) Determine whether the point C with coordinate (2, 0, -1) lies on this line.

3-10-1-ib-aa-hl-vector-equation-of-a-line-we-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.