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

    • $y=mx+c$

    • 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 $i,j,k$reference unit vectors or as column vectors 

• It follows that in a vector equation of a line either form can be employed – for example,

 $r=3i+j-7k+t\left(i-2j\right)$  and  $r=\left(\begin{array}{c}3\\ 1\\ -7\end{array}\right)+t\left(\begin{array}{c}1\\ -2\\ 0\end{array}\right)$   

                             show the same equation written using the two different forms

How do I know 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 vector 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

  • $x-2y=5$ and $3x-6y=15$

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

  • $\mathbf{r}=\left(\begin{array}{c}5\\ 0\end{array}\right)+t\left(\begin{array}{c}2\\ 1\end{array}\right)$ and $\mathbf{r}=\left(\begin{array}{c}1\\ -2\end{array}\right)+t\left(\begin{array}{c}-2\\ -1\end{array}\right)$