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
- r = a + t (b - a)
- 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
- For example: if an equation for a line is r = 3i + 2j - k + t (i + 2j)
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
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
b)
Determine whether the point C with coordinate (2, 0, -1) lies on this line.