Using determinants prove that the points `(a ,\ b),\ (a^(prime),\ b^(prime))a n d\ (a-a^(prime),\ b-b^(prime))` are collinear if `a b^(prime)=a^(prime)bdot`

Explanation

We have to prove that the points `(a ,\ b),\ (a^(prime),\ b^(prime))a n d\ (a-a^(prime),\ b-b^(prime))`
are collinear if `a b^(prime)=a^(prime)b`

Now any three points are collinear if the area of the triangle formed by these points will be zero.

As we know if `A(x_1, y_1),B(x_2, y_2)a n dC(x_3,y_3)`
are vertices of a triangle

Then the area of triangle are given as:

`Delta=1/2[(x_1,y_1, 1),(x_2,y_2, 1),(x_3,y_3 ,1)]=0`

`1/2[(a,b, 1),(a',b', 1),(a-a',b-b' ,1)]=0`

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