Question

If A(3, 2, -1), B(-2, 2, -3), C(3, 5, -2), D(-2, 5, -4) then find its area.

Answer 1

`bar"AB" = bar"b" - bar"a" = (- 2hat"i" + 2hat"j" - 3hat"k") - (3hat"i" + 2hat"j" - hat"k")`

`= - 5hat"i" - 2hat"k"`

`bar"AD" = bar"d" - bar"a" = (- 2hat"i" + 5hat"j" - 4hat"k") - (3hat"i" + 2hat"j" - hat"k")`

`= - 5hat"i" + 3hat"j" - 3hat"k"`

∴ `bar"AB" xx bar"AD" = |(hat"i",hat"j",hat"k"),(-5,0,-2),(-5,3,-3)|`

`= (0 + 6)hat"i" - (15 - 10)hat"j" + (- 15 + 0)hat"k"`

`= 6hat"i" - 5hat"j" - 15hat"k"`

∴ area of parallelogram = `|bar"AB" xx bar"AD"|`

`= sqrt(6^2 + (- 5)^2 + (- 15)^2)`

`= sqrt(36 + 25 + 225)`

`= sqrt286` sq units.