PQRS is a parallelogram, where `vec(PQ)=3hat(i)+2hat(j)-mhat(k), vec(PS)=hat(i)+3hat(j)+hat(k)` and the area of the parallelogram is `sqrt(90)`. What is the value of m?

Options

A. `1`
B. `-1`
C. `2`
D. `-2`

Answer

A

Explanation

Let `vec(PQ)=3hat(i)+2hat(j)-m hat(k) and vec(PS)=hat(i)+3hat(j)+hat(k)`
where PQRS is a parallelogram.
`:.` Area of parallelogram `=||(hat(i), hat(j), hat(k)),(3, 2, -m),(1,3,1)||`
`=|hat(i)(2+3m)-hat(j)(3+m)+hat(k)(9-2)|`
`=sqrt((2+3m)^(2)+(3+m)^(2)+7^(2))`
`rArr 90=4+9m^(2)+12m+9+m^(2)+6m+49`
`rArr 10m^(2)+18m-28=0`
`rArr 5m^(2)+9m-14=0`
`rArr 5m^(2)+14m-5m-14=0`
`rArr m(5m+14)-1(5m+14)=0`
`rArr (5m+14)(m-1)=0`
`rArr m=1 or (-14)/(5)`