Edexcel A Level Further Maths: Core Pure:复习笔记6.1.1 Equations of Lines in 3D

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

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
  • For vector equations this is not true – two equations might look different but still represent the same line:

Exam 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


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



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  then  becomes
      • Where  is a position vector and  is a direction vector
    • This vector equation can then be split into its three separate component forms:

Worked Example

Write the parametric form of the equation of the line which passes through the point (-2, 1, 0) with direction vector .


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 λ from the parametric equations
      • λ can be eliminated by making it the subject of each of the parametric equations
      • For example: gives
  • In 2D the cartesian equation of a line is a regular equation of a straight line simply given in the form
    • by rearranging
  • In 3D the cartesian equation of a line also includes z and is given in the form
    • where
    • This is given in the formula booklet
  • If one of your variables does not depend on λ then this part can be written as a separate equation
    • For example: gives

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
  • A vector equation of the line can be found by
    • STEP 1: Set each part of the equation equal to λindividually
    • 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
    • STEP 3: Write this in the vector form
    • STEP 4: Set r  to equal
  • If one part of the cartesian equation is given separately and is not in terms of λ then the corresponding component in the direction vector is equal to zero

Worked Example

A line has the vector equation . Find the Cartesian equation of the line.