Vector line equation

As well as being able to represent a line in cartesian form (e.g. ), we can also represent a line using vectors.

Equation of a line in vector form

Finding the vectors between points

The first step to finding the equation of a line using vectors is to find the vectors between all points.

Find the vector between the point and

• Answer:

Find ALL vectors between the points , and

• Answers:

Finding the vector equation from points

Find a vector equation of the line between the points and

• Let be the position vector of from the origin:
• So the direction vector is .
• Equation of a line:
• Answer:

Note: there are multiple solutions to this line equation from the points given. We can find other ones by using different vectors (e.g. using instead of ).

Find a vector equation of the line between the points and

• Let be the position vector of from the origin:
• So the direction vector is , which we can simplify to (because it’s just a direction and the magnitude is not important here).
• Equation of a line:
• Answer:

Note: there are multiple solutions to this line equation from the points given. We can find other ones by using different vectors (e.g. using instead of ).

Find a vector equation of the line between the points and

• Let be the position vector of from the origin:
• So the direction vector is , which we can simplify to (because it’s just a direction and the magnitude is not important here).
• Equation of a line:
• Answer:

More complex exam-style questions

is the point and . Find if and are on the line

• Find the length of vector :
• Find :
  • or
• Answer: or