Question

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is `1/2 |bar"AC" xx bar"BD"|`

Answer 1

Vector area of quadrilateral ABCD = {Vector area of ∆ABC} + {Vector area of ∆ACD}

= `1/2 (bar"AB" xx bar"AC") + 1/2(bar"AC" xx bar"AD")`

= `- 1/2 (bar"AC" xx bar"AB") + 1/2(bar"AC" xx bar"AD")`

= `1/2bar"AC" xx [- bar"AB" + bar"AD"]`

= `1/2bar"AC" xx [bar"BA" + bar"AD"]`

= `1/2 bar"AC" xx bar"BD"`

∴ The area of the quadrilateral ABCD = `1/2|bar"AC" xx bar"BD"|`