There are two useful versions of the vector equation of a line, and the one we choose depends upon what information is given:

1. The vector equation of a line can be obtained by using a fixed point on the line and a vector parallel to the line.
2. The vector equation of a line can be obtained by using the position vectors of two points A and B on the line.

Case 1. Given a fixed point on the line and a vector parallel to the line.
Suppose the fixed point on the line is whose position vector with respect to the origin is OP and the vector with the same direction of the line is . Let be any point on the line and let the position vector of be .

From the diagram we can see that .
And we know that has the same direction as , and therefore for some real number .
So we have: (the vector equation of a line).

Case 2. Given two points on the line.
Suppose the two given points are A and B. Then find the vector AB and return to case 1.

Vector equation of a line

Find the vector equation of the line which is parellel to and passes through the point

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Find the vector equation of the line passing through the points P=(-2,3) and Q=(1,4)

Firstly, we find the vector PQ: .
The vector equation is:

Find the vector equation of the line that passes through the points and

=( , ) ( , )