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 ABC'D' be two parallelograms on the same base AB and between the same parallels l and m

The vector area of parallelogram

$\begin{array}{c}ABCD=\overrightarrow{AB}\times \overrightarrow{AD}\\ =\overrightarrow{AB}\times \left(\overrightarrow{A{D}^{\prime }}+\overrightarrow{D^{\prime }D}\right)\\ =\overrightarrow{AB}\times \overrightarrow{A{D}^{\prime }}+\overrightarrow{AB}\times \overrightarrow{D^{\prime }D}\\ =\overrightarrow{AB}\times \overrightarrow{A{D}^{\prime }}+0\left(\overrightarrow{AB}and\overrightarrow{D^{\prime }D}areparallel\right)\\ =\overrightarrow{AB}\times \overrightarrow{A{D}^{\prime }}\end{array}$

=Vector area of parallelogram ABC'D'. Hence proved