Example 4: Determining Whether Tree Points Are on the Same Line

Determine whether the three points

A(1,3), B(2,5), C(3,10)

are on the same line.

The line L through A and B has the vector equation

X = A + t(B - A)

= i + 3j + t(i + 2j) = (1 + t)i + (3 + 2t)j.

The only point on L with x component 3 is given by

3 = 1 + t, t = 2, P = 3i + 7j.

Since C is another point with x component 3, C is not on L. Therefore A, B, and C are not on the same line, as we see in Figure 10.2.10.

Figure 10.2.10