The equation of the line joining points and can be written in parametric form. Let the coordinates of a point on the line be . The parametric equations for the line are: Now, the coordinates of the foot of the perpendicular from on the line will satisfy the condition that the vector from to the point on the line is perpendicular to the direction vector of the line. The direction vector of the line joining and is . The vector from to a point on the line is: The dot product of this vector with the direction vector must be zero for the vectors to be perpendicular. Therefore: Expanding this:
Substituting into the parametric equations of the line to find the coordinates of the foot of the perpendicular: Thus, the coordinates of the foot of the perpendicular are .