INTRODUCTION

Determinats can be used to find equations of lines, curves and surfaces passing through the given fixed points. The following are few examples of this application:

Suppose that two points A = (x,y) and B= (x', y') in th XY-plane are given. Find the equation the line passing through these two points.

If three points A, B and C are given, find the equation of the circle passing therough these three points?

If you know that by Keplers first law the orbit of an asteroid about the sun is an ellipse, how many observation of the asteroid's position do you need to find the equation of the orbit?

based on the knowledge from your previous math courses ( including linear algebra ) you should be able to answer the following questions:

How many points do you need to find the equation? What if you do not have the exact number of points? What if you have more than needed points( See application 10, least square approximation )

In three dimentional space, How many points are needed to to define a plane? How can you find the equation of the plane passing through 3-points? How about finding the equation of a sphere passing through four given points?

We can use linear algebra to find the equation of a curve or a surface whose equation ( with unknown coefficents ) is given. For example to find the equation of the line passing through points A and B in the plane, we need to find the coefficients "a" , "b" and "c" so that the points A and B satisfy the equation ax+by +c=0.

The basic idea is to sustitute the coordinates of the given points ( n-1 given points and one general point as (x, y) ) in the equation and get a linear system whose variables are the coefficients of the equation. This will yield a linear system of n-equation and n-unknowns. Instead of solving this system we may use the fact that a homogeneous linear system has a non-trivial solution if and only if determinant of the coefficient matrix is zero. Therefore etting determinat of the coeeficient matrix equal to zero, will give the equation.

Note that in an equation with n-unknowns you can divide through by one of the non-zero coefficients to reduce it to (n-1) coefficient.