Problem

Solution

The vectorial equation of a line is the equation satisfied by the position vector of any point of this line; in this case we want to express that equation in terms of the position vectors of the points A and B. The parametric equation is obtained from the vectorial equation with the coefficients of the unit vectors.

1. To construct the vectorial equation of the line, let  and  be vectors pointing to A and B. The line we want to represent with a vectorial equation is then parallel to the vector difference

Now let  be the position vector of any point of this line. Then the vector  is also parallel to the line and so to ; therefore

This is the vectorial equation of the line passing through A and B. The position vector  of any point belonging to this line satisfies this equation.

2. To construct the parametric representation for this line, turn the generic points A and B into concrete points; that is, give values to the components of  and .  For example:

Regardless of the values of  and , for the position vector of a point of the line, we always have

Thus the vectorial equation for these particular points A and B becomes

Since the unit vectors [] are independent, this vectorial equation represents three equations, constituted by the coefficients of these unit vectors

This is a system of three equations in terms of three unknowns x<