Equations of Lines in 3D (Edexcel A Level Further Maths: Core Pure)

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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 bold i comma bold space bold j comma bold space bold k reference unit vectors or as column vectors 
  • It follows that in a vector equation of a line either form can be employed – for example,

 bold r equals 3 bold i plus bold j minus 7 bold k plus t open parentheses bold i minus 2 bold j close parentheses  and  bold 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

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!

Worked example

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

~YvxQzGe_picture-1

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

al-fm-6-1-1-vector-equation-of-line-we-solution-b

Equation of a Line in Parametric Form

How do I find the vector equation of a line in parametric form?

  • By considering the three separate components of a vector in the x, y and z directions it is possible to write the vector equation of a line as three separate equations
    • Letting bold italic r equals blank open parentheses fraction numerator x over denominator table row y row z end table end fraction close parentheses then bold italic r equals bold italic a plus lambda bold italic b becomes
    • open parentheses fraction numerator x over denominator table row y row z end table end fraction close parentheses 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
      • Where 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 is a position vector and begin mathsize 16px style 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 end style is a direction vector
    • This vector equation can then be split into its three separate component forms:
      • x equals blank a subscript 1 plus blank lambda b subscript 1 blank
      • y equals blank a subscript 2 plus blank lambda b subscript 2 blank
      • z equals blank a subscript 3 plus blank lambda b subscript 3 blank

Worked example

Write the parametric form of the equation of the line which passes through the point (-2, 1, 0) with direction vector open parentheses fraction numerator 3 over denominator table row 1 row cell negative 4 end cell end table end fraction close parentheses.

al-fm-6-1-1-parametric-equation-of-line-we-solution

Equation of a Line in Cartesian Form

  • The Cartesian equation of a line can be found from the vector equation of a line by
    • Finding the vector equation of the line in parametric form
    • Eliminating lambda from the parametric equations
      • lambda can be eliminated by making it the subject of each of the parametric equations
      • For example: blank x equals blank x subscript 0 plus blank lambda l blankgives blank lambda equals blank fraction numerator blank x minus blank x subscript 0 over denominator l end fraction blank
  • In 2D the cartesian equation of a line is a regular equation of a straight line simply given in the form
    •  y equals m x plus c
    • a x plus b y plus d equals 0
    • fraction numerator y minus y subscript 1 over denominator y subscript 2 minus y subscript 1 end fraction equals fraction numerator x minus x subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction by rearranging y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses
  • In 3D the cartesian equation of a line also includes z and is given in the form
    • fraction numerator x minus blank a subscript 1 over denominator b subscript 1 end fraction equals blank fraction numerator y minus blank a subscript 2 over denominator b subscript 2 end fraction equals blank fraction numerator z minus blank a subscript 3 over denominator b subscript 3 end fraction left parenthesis equals lambda right parenthesis
    • where open parentheses fraction numerator x over denominator table row y row z end table end fraction close parentheses 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
    • This is given in the formula booklet
  • If one of your variables does not depend on lambda then this part can be written as a separate equation
    • For example: b subscript 2 equals 0 blank rightwards double arrow y equals blank a subscript 2 blankgives fraction numerator x minus blank a subscript 1 over denominator b subscript 1 end fraction equals blank fraction numerator z minus blank a subscript 3 over denominator b subscript 3 end fraction comma blank y equals blank a subscript 2

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

  • If you are given the Cartesian equation of a line in the form
    • fraction numerator x minus blank a subscript 1 over denominator b subscript 1 end fraction equals blank fraction numerator y minus blank a subscript 2 over denominator b subscript 2 end fraction equals blank fraction numerator z minus blank a subscript 3 over denominator b subscript 3 end fraction left parenthesis equals lambda right parenthesis
  • A vector equation of the line can be found by
    • STEP 1: Set each part of the equation equal to lambdaindividually
    • STEP 2: Rearrange each of these three equations (or two if working in 2D) to make x, y, and z the subjects
      • This will give you the three parametric equations
      • x equals blank a subscript 1 plus blank lambda b subscript 1 blank
      • y equals blank a subscript 2 plus blank lambda b subscript 2 blank
      • z equals a subscript 3 plus blank lambda b subscript 3 blank
    • STEP 3: Write this in the vector form stretchy left parenthesis fraction numerator x over denominator table row y row z end table end fraction stretchy right parenthesis equals blank stretchy left parenthesis 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 stretchy right parenthesis plus lambda stretchy left parenthesis 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 stretchy right parenthesis
    • STEP 4: Set r  to equal open parentheses fraction numerator x over denominator table row y row z end table end fraction close parentheses
  • If one part of the cartesian equation is given separately and is not in terms of lambda then the corresponding component in the direction vector is equal to zero

Worked example

A line has the vector equation r blank equals blank open parentheses table row 1 row 0 row 2 end table close parentheses plus lambda open parentheses table row 4 row cell negative 2 end cell row 1 end table close parentheses. Find the Cartesian equation of the line.

al-fm-6-1-1-cartesian-equation-of-line-we-solution-a

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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.