Question

Find the area of a parallelogram whose adjacent sides are given by the vectors:

`veca = veci − vecj + 3veck and vecb = 2veci − 7vecj + 4veck`

Answer 1

The area of a parallelogram with adjacent sides and is given by:

Area = `|veca × vecb|`

Cross Product Calculation

Calculation `veca × vecb` using the determinant method:

`veca × vecb = |(hati,hatj,hatk),(1,−1,3),(2,−7,4)|`

Expanding along the first row:

`hati` component: (−1) (4) − (−7) (3)

 = −4 + 21

= 17

`hati` component: −[(1) (4) − (2) (3)

= −(4 − 6)

= 2

`hatk` component: (1) (−7) − (2) (−1)

= −7 + 2 

= −5

so, `veca × vecb = 17hati + 2j − 5hatk`

Magnitude of the Area:

Area = `sqrt((17)^2 + (2)^2 + (−5)^2)`

Area = `sqrt(289 + 4 + 25)`

Area = `sqrt318` sq.units