Relative to a fixed origin O, the point A has position vector (8i+13j-2k), the point B has position vector (10i+14j-4k). A line l passes through points A and B. Find the vector equation of this line.

The vector equation of any line is composed of the addition of a point on the line and some distance in the direction of the line, in the form y=(x,y,z) +d(x_1, y_1, z_1), where (x,y,z) is any point on the line, d is some scalar magnitude and (x_1,_1,z_1) is the direction of the line (this can be represented on a drawing that will help visualisation). For the point in the line we can either pick A or B, it doesn't matter which one. For the direction, we want to find the vector from A to B and for this we must do AO (A to the origin) and add it to OB (origin to B). Therefore we do -(8i+13j-2k)+(10i+14j-4k)=(2i+j-2k). AO will the negative position vector of A because AO is going in the opposite direction to the position vector which goes from O to A. We can now write the vector equation of the line as y=(8,13,-2)+d(2,1,-2).