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A LevelMaths: Pure 3CIERevision Notes7. Vectors7.3 Further Vectors7.3.1 Equation of a Line in Vector Form

Equation of a Line in Vector Form (CIE A Level Maths: Pure 3)

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Equation of a Line in Vector Form

How do I find the vector equation of a line?

  • You need to know:
    • The position vector of one point on the line
    • A direction vector of the line (or the position vector of another point)
  • There are two formulas for getting a vector equation of a line:
    • r = a + t (b - a)
      • use this formula when you know the position vectors a and b of two points on the line
    • r = a + t d
      • use this formula when you know the position vector a of a point on the line and a direction vector d
    • Both forms could be compared to the Cartesian equation of a 2D line
      • y space equals space m x space plus space c
      • The point on the line a is similar to the “+c” part
      • The direction vector d or b – a is similar to the “m” part
  • The vector equation of a line shown above can be applied equally well to vectors in 2 dimensions and to vectors in 3 dimensions
  • Recall that vectors may be written using i comma space j comma space kreference unit vectors or as column vectors 
  • It follows that in a vector equation of a line either form can be employed – for example,

 r equals 3 i plus j minus 7 k plus t open parentheses i minus 2 j close parentheses  and  r equals open parentheses table row 3 row 1 row cell negative 7 end cell end table close parentheses plus t open parentheses table row 1 row cell negative 2 end cell row 0 end table close parentheses   

                             show the same equation written using the two different forms

How do I determine if a point is on a line?

  • Each different point on the line corresponds to a different value of t
    • For example: if an equation for a line is r = 3i + 2j - k + t (i + 2j)
      • the point with coordinates (2, 0, -1) is on the line and corresponds to t = -1
    • However we know that the point with coordinates (-7, 5, 0) is not on this line
      • No value of t could make the k component 0

 

Can two different equations represent the same line?

  • Why do we say a direction vector and not the direction vector? Because the magnitude of the vector doesn’t matter; only the direction is important
    • we can multiply any direction vector by a (non-zero) constant and this wouldn’t change the direction
  • Therefore there are an infinite number of options for a (a point on the line) and an infinite number of options for the direction vector
  • For Cartesian equations – two equations will represent the same line only if they are multiples of each other
    • x space minus space 2 y space equals space 5 and 3 x space minus space 6 y space equals space 15
  • For vector equations this is not true – two equations might look different but still represent the same line:
    • bold r equals open parentheses table row 5 row 0 end table close parentheses plus t open parentheses table row 2 row 1 end table close parentheses and bold r equals open parentheses table row 1 row cell negative 2 end cell end table close parentheses plus t open parentheses table row cell negative 2 end cell row cell negative 1 end cell end table close parentheses

Worked example

7-3-1-equation-of-a-line-in-vector-form-we-solution-part-1

7-3-1-equation-of-a-line-in-vector-form-we-solution-part-2

Examiner Tip

  • Remember that the vector equation of a line can take many different forms. This means that the answer you derive might look different from the answer in a mark scheme. 
  • You can choose whether to write your vector equations of lines using reference unit vectors or as column vectors – use the form that you prefer!
  • If, for example, an exam question uses column vectors, then it is usual to leave the answer in column vectors, but it isn’t essential to do so - you’ll still get the marks!

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    Previous:7.2.2 Problem Solving using 3D VectorsNext:7.3.2 Parallel, Intersecting & Skew Lines
    Amber

    Author: Amber

    Expertise: Maths

    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.