Equations of Lines in 3D (Edexcel A Level Further Maths) : Revision Note

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,

   and     

                             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

  • and 

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

  • and 

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:

    • 

    • 

    • 

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  from the parametric equations

    •  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  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 individually

  • 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 then the corresponding component in the direction vector is equal to zero