Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.

Explanation

Let ABCD and ABFE are parallelograms on the same base AB and between the same parallel lines AB and DF.
Here, AB||CD and AE||BF

Let `vec(AB)=vecaandvec(AD)=vecb` :. Area of parallelogram `ABCD=vecaxxvecb`
Now, area of parallelogram `ABFE=vec(AB)xxvec(AE)`
`=vec(AB)xx(vec(AD)+vec(DE))`
`=vec(AB)xx(vecb+kveca)" "["let"vec(DE)=kveca,"where k is a scalar"]`
`=vecaxx(veca+kveca)`
`=(vecaxxvecb)+(vecaxxkveca)`
`=(vecaxxvecb)+k(veca+veca)`
`=(vecaxxvecb)" "[:.vecaxxveca=0]`
= Area of parallelogram ABCD
Hence proved.