Question

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

Answer 1

Let ABCD and ABFE be two parallelograms on the same base AB and between same parallel lines AB and DF.

Let `vec"AB" = vec"a"` and `vec"AD" = vec"b"`

∴ Area of parallelogram ABCD = `|vec"a" xx vec"b"|`

= `|vec"a" xx (vec"AD" + vec"DE")|`

= `|vec"a" xx (vec"b" xx "K"vec"a")|`

= `|(vec"a" xx vec"b") + "K"(vec"a" xx vec"a")`

= `|vec"a" xx vec"b"| + 0`   ...`["becuase"  vec"a" xx vec"a" = 0]`

= `|vec"a" xx vec"b"|`

Hence proved.