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

Explanation

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 edges are `veca , vecb and vecc is [ veca vecb vecc] = veca * ( vecb xx vecc).`
`vec(AB) = (2 - 1) hat i + (1 -1) hatj + (3 - 1) hat k`
` = hati + 2 hatk`
` vec (AC) = (3 -1) hati + (2 -1) hatj + (2 -1) hatk`
` = 2 hati + hatj + hatk`
` vec(AD) = (3 -1) hati + (3 -1) hatj + (4-1) hatk`
` = 2 hati + 2 hatj + 3 hatk`
` [veca vecb vecc] = |(1,0,2),(2,1,1),(2,2,3)|`
` = 1 (3-2) - 0 + 2 (4-2)`
` = 1 + 4`
= 5 cubic units