# Consider the four points A(2, 4, 1), B(3,-1, 2), C(-1, 1, 1) and D(6, 2, 2). Is ABCD a parallelogram? Find the area of ABCD.

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### 1 Answer

The line `AB` contains the point `A:(2,4,1)` and has the directional vector `<AB> =<3-2,-1-4,2-1> =<1,-5,1>` .

The parametric equation of line `AB` is

`(x,y,z) =(2,4,1)+t*<1,5,1>`

Similar for the lines `BC` ,`CD` and `DA` the equations are

`BC: (x,y,z) =(3,-1,2) +t*<-4,2,-1>`

`CD: (x,y,z) =(-1,1,1) +t*<7,1,1>`

`DA: (x,y,z) =(6,2,2) +t*<-4,2,-1>`

The lines `BC||DA` since they have the same directional vector.

The line `AB` is not parallel to `CD` since

`<AB>xx<CD> =<1,-5,1>xx<7,1,1> = |[i,j,k],[1,-5,1],[7,1,1]| =-6i-8j-34k =<-6,-8,-34> !=0` This figure obtained is below attached.

**Answer: The figure ABCD is not a parallelogram**.