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
Examiner Tips and Tricks
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
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b) Determine whether the point C with coordinate (2, 0, -1) lies on this line.
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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 .
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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
Worked Example
A line has the vector equation . Find the Cartesian equation of the line.
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