7.3.1 Equation of a Line in Vector Form

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
  - $y = m x + c$
  - 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,

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

show the same equation written using the two different forms

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

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
- $x - 2 y = 5$ and $3 x - 6 y = 15$
For vector equations this is not true – two equations might look different but still represent the same line:
- $r = (\begin{matrix} 5 \\ 0 \end{matrix}) + t (\begin{matrix} 2 \\ 1 \end{matrix})$ and $r = (\begin{matrix} 1 \\ - 2 \end{matrix}) + t (\begin{matrix} - 2 \\ - 1 \end{matrix})$

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

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!

CIE A Level Maths: Pure 3

7.3.1 Equation of a Line in Vector Form