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
- First check that they are parallel
- Sometimes the vector equations of the lines may look different