Edexcel A Level Further Maths: Core Pure:復習筆記6.1.1 Equations of Lines in 3D
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,
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
- For vector equations this is not true – two equations might look different but still represent the same line:
Exam 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.
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
- Where
- This vector equation can then be split into its three separate component forms:
- Letting
Worked Example
Write the parametric form of the equation of the line which passes through the point (-2, 1, 0) with direction vector 
.
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 λ?from the parametric equations
- λ 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 λ?then this part can be written as a separate equation
- For example:
gives 
- For example:
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 λindividually
- 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 λ?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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