Using vectors, find the area of the `DeltaABC` with vertices A(1, 2, 3), B(2, -1, 4) and C(4, 5, -1)

Explanation

Here, `vec(AB)=(2-1)hati+(-1-2)hatj+(4-3)hatk`
`=hati-3hatj+hatk`
and `vec(AC)=(4-1)hati+(5-2)hatj+(-1-3)hatk`
`3hati+3hatj-4hatk`

`:.vec(AB)xxvec(AC)|[hati,hatj,hatk],[1,-3,1],[3,3,-4]|`
`=hati(12-3)-hatj(-4-3)hatk(3+9)`
`=9hati+7hatj+12hatk`
and `|vec(AB)xxvec(AC)|=sqrt(9^(2)+7^(2)+12^(2))`
`=sqrt(81+49+144)`
`sqrt274`
:. Area of `DeltaABC=(1)/(2)|vec(AB)xxvec(AC)|`
`(1)/(2)sqrt274` sq units