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