# Edexcel A Level Further Maths: Core Pure:复习笔记6.1.2 Pairs of Lines in 3D

### Coincident & Parallel Lines

#### How do I tell if two lines are parallel?

• Two lines are parallel if, and only if, their direction vectors are parallel
• This means the direction vectors will be scalar multiples of each other
• For example, the lines whose equations are and  are parallel
• This is because

#### How do I tell if two lines are coincident?

• Coincident lines are two lines that lie directly on top of each other
• They are indistinguishable from each other
• Two parallel lines will either never intersect or they are coincident (identical)
• Sometimes the vector equations of the lines may look different
• for example, the lines represented by the equations  and  are coincident,
• To check whether two lines are coincident:
• First check that they are parallel
• They are because  and so their direction vectors are parallel
• Next, determine whether any point on one of the lines also lies on the other
• is the position vector of a point on the first line and  so it also lies on the second line
• If two parallel lines share any point, then they share all points and are coincident

### Intersecting Lines

#### How do I determine whether two lines in 3 dimensions intersect?

• If the lines are not parallel, check whether they intersect:
• STEP 1: Set the vector equations of the two lines equal to each other with different variables
• e.g. λ and μ, for the parameters
• STEP 2: Write the three separate equations for the i, j, and k components in terms of λ and μ
• STEP 3: Solve two of the equations to find a value for λ and μ
• STEP 4: Check whether the values of λ and μ you have found satisfy the third equation
• If all three equations are satisfied, then the lines intersect

#### How do I find the point of intersection of two lines?

• If a pair of lines are not parallel and do intersect, a unique point of intersection can be found
• If the two lines intersect, there will be a single point that will lie on both lines
• Follow the steps above to find the values of λ and μ that satisfy all three equations
• STEP 5: Substitute either the value of λ or the value of μ into one of the vector equations to find the position vector of the point where the lines intersect
• It is always a good idea to check in the other equations as well, you should get the same point for each line

#### Exam Tip

• Make sure that you use different letters, e.g. λ and μ, to represent the parameters in vector equations of different lines
• Check that the variable you are using has not already been used in the question

#### Worked Example

Line L1 has vector equation .

Line L2 has vector equation .

a) Show that the lines L1 and L2 intersect.

b) Find the position vector of the point of intersection.

### Skew Lines

#### What are skew lines?

• Lines that are not parallel and which do not intersect are called skew lines
• This is only possible in 3-dimensions
• If two lines are skew then there is not a plane in 3D than contains both of the lines

#### How do I determine whether lines in 3 dimensions are parallel, skew, or intersecting?

• First, look to see if the direction vectors are parallel:
• if the direction vectors are parallel, then the lines are parallel
• if the direction vectors are not parallel, the lines are not parallel
• If the lines are parallel, check to see if the lines are coincident:
• If they share any point, then they are coincident
• If any point on one line is not on the other line, then the lines are not coincident
• If the lines are not parallel, check whether they intersect:
• STEP 1: Set the vector equations of the two lines equal to each other with different variables
• e.g. λ and μ, for the parameters
• STEP 2: Write the three separate equations for the i, j, and k components in terms of λ and μ
• STEP 3: Solve two of the equations to find a value for λ and μ
• STEP 4: Check whether the values of λ and μ you have found satisfy the third equation
• If all three equations are satisfied, then the lines intersect
• If not all three equations are satisfied, then the lines are skew

#### Worked Example

Determine whether the following pair of lines are parallel, intersect, or are skew.

.