Question

The two adjacent sides of a parallelogram are `2hati-4hatj-5hatk and 2 hati+2hatj+3hatj` . Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.

Answer 1

The two adjacent sides of a parallelogram are ` 2hati-4hatj-5hatk and 2 hati+2hatj+3hatj`

suppose `veca=2hati-4hatj-5hatk and vecb=2 hati+2hatj+3hatj`

Then any one diagonal of a parallelogram is `vecP=veca+vecb`

`vecP=veca+vecb`

`=2hati-4hatj-5hatk + 2 hati+2hatj+3hatj`

`=4hati-2hatj-2hatk`

Therefore, unit vector along the diagonal is `vecp/|vecp|=(4hati-2hatj-2hatk)/sqrt(16+4+4)=(2hati-hatj-hatk)/sqrt(6)`

Another diagonal of a parallelogram is `vecP=vecb-veca`

`vecP=vecb-veca`

`=2hati+2hatj+3hatk-2hati+4hatj+5hatk`

`=6hatj+8hatk`

Therefore, unit vector along the diagonal is `vecp/|vecp|=(6hatj+8hatk)/sqrt(36+64)=(6hatj+8hatk)/sqrt(10)=(3hatj+4hatk)/5`

Now

`vecPxxvecP=|[hati,hatj,hatk],[4,-2,-2],[0,6,8]|`

`=hati(-16+12)-hatj(32-0)+hatk(24-0)`

`=-4hati-32hatj+24hatk`

Area of parallelogram= `|vecpxxvecp|/2=sqrt(16+1024+576)/2=sqrt(1616)/2=4sqrt(101)/2=2sqrt101 `