What is the area of the parallelogram having diagonals `3hat(i) + hat(j) - 2 hat(k) and hat(i) - 3 hat(j) + 4 hat(k)`?

Options

A. `5sqrt(5)` square units
B. `4sqrt(5)` square units
C. `5sqrt(3)` square units
D. `15sqrt(2)` square units

Answer

C

Explanation

Diagonal `d_(1), vec(AC) = 3i + j -2k`
Diagonal `d_(2), vec(BD)= I -3 j + 4k`

Area of parallelogram is `(1)/(2)|vec(d_(1))xxvec(d_(2))|`
Hence area `=(1)/(2)|(hat(i),hat(j),hat(k)),(3,1,-2),(1,-3,4)|`
`=(1)/(2)|[hat(i)(4-6)-hat(j)(12+2)+hat(k)(-9-1)]|`
`(1)/(2)|-2hat(i) - 14 hat(14)-10hat(k)|`
`=(1)/(2)sqrt(4+196+100)`
`=(10sqrt(3))/(2)=5sqrt(3)` square units