If the two diagonals of a parallelogram are given by `vecR_1=3hati-2hatj+7hatk and vecR_2=5hati+6hatj-3hatk` , find out the area of this parallelogram.

Explanation

Let the two adjacent sides of the parallelogram be `vecA and vecB`. Then,
`vecR_1=vecA+vecB and vecR_2=vecA-vecB` [Fig.2.61]

Now, `(vecA+vecB)xx(vecA-vecB)=vecAxxvecA-vecAxxvecB+vecBxxvecA-vecBxxvecB`
`=0-vecAxxvecB-vecAxxvecB-0 =-2(vecAxxvecB)`
From this given data,
`(vecA+vecB)xx(vecA-vecB)=vecR_1xxvecR_2`
`=(3hati-2hatj+7hatk)xx(5hati+6hatj-3hatk)`
`=|{:(hati," "hatj," "hatk),(3,-2," "7),(5," "6,-3):}|= hati[(-2)(-3)-7xx6]+hatj[7xx5-3(-3)]+hatk[3xx6-5(-2)]`
`= -36 hatj +44hatj+28hatk`
On comparison , `-2(vecAxxvecB)=-36hati+44hatj+28hatk`
`or, vecAxxvecB=18hatj-22hatj-14hatk`
Here, `vecAxxvecB=vecS` = area vector for the parallelogram.
`therefore` its area `|vecS|=|vecAxxvecB|=sqrt(18^2+(-22)^2+(-14)^2)=31.7` square units.