Question

If A, B, C, D are (1, 1, 1), (2, I, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.

Answer 1

Given that A,B,C and D are (1, 1, 1), (2, 1, 3) , (3, 2, 2) and (3, 3, 4) respectively.
We need to find the volume of the parallelopiped with AB, AC and AD as the concurrent edges.
The volume of the parallelopiped whose coterminus edges are a, b and c is `[veca vecbvecc]=veca.(vecbxxvecc)`

Given that A,B,C and D are (1, 1, 1) , (2, 1, 3) , (3, 2, 2) and (3, 3, 4)

`vec(AB)=(2-1)hati+(1-1)hatj+(3-1)hat k`

`=hati+2hatk`

`vec(AC)=(3-1)hati+(2-1)hatj+(2-1)hat k`

`=2hati+hatj+hatk`

`vec(AD)=(3-1)hati+(3-1)hatj+(4-1)hat k`

`=2hati+2hatj+3hatk`

`[veca vecbvecc]=|[1,0,2],[2,1,1],[2,2,3]|`

                =1(3-2)-0+2(4-2)

                =1+4

                = 5 cubic units