Question

Why area of triangle by determinants method contain 1 in last column

Answer 1

Answer:

Because it's a special case of a more general 3 dimensional concept.

In a sense, order 2 determinants correspond to areas, order 3 to volumes, and so on.  But here, we're using an order 3 determinant for an area... what's going on?

Basically, think of the plane that you're working with as the plane z=1 in 3 dimensional space.  So the points (a,b), (c,d), (e,f) that make up the vertices of the triangle are really (a,b,1), (c,d,1), (e,f,1) in 3-space.  The determinant made up from these three points corresponds to the volume of a parallelepiped that is determined by these three points and the origin at (0,0,0).  This in turn is related to the area of the triangle that we want because it is a cross section of the parallelepiped through three vertices.

That's the basic idea anyway.  Hope that helps.

Answer 2

Since the area is a positive quantity,we always take the absolute value of the determinate in (1).if the area is given.uses both positive and negitive value of the determinat for caculation.the area of the tringle formed by three collinear points is 0.