# Edexcel A Level Further Maths: Core Pure:复习笔记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
• 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

#### 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
• 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 .

### 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

#### 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.