If `a=hat(i)+hat(j)+hat(k), b=hat(i)+3hat(j)+5hat(k)` and `c=7hat(i)+9hat(j)+11 hat(k)`, then the area of the parallelogram having diagonals `a+b` and `b+c` is

Options

A. `4sqrt(6)`
B. `1/2 sqrt(21)`
C. `sqrt(6)/2`
D. `sqrt(6)`

Answer

B

Explanation

Given that,
`a = hati +hatj +hatk, b =hati +3hatj +5 hatj `
and `c=7 hati +9 hatj +1 hatk`
Let `A =1 +b=(hati + hatj +hatk) +(hati+3hatj +5 hatk)=2 hati +4 hatj+6hatk`
and `=+=(hat+hat3+ha5)+(hat7+hat9+hat11)`
`=hat8+hat12+hat16`
If A and B are adagonals, then area of parallelogram
`1/2|AxxB|=1/2||{:(hati, hatj,hatk),(2,4,6),(8,12,16):}||`
`=1/2|hati(64-72)-hatj(32-48)+hatk(24-32)|`
`1/2|-8hati+16hatj-8hatk|`
`=|-4hati+8hatj+4hatk|`
`=sqrt((-4)^(2)+(8)^(2)+(-4)^(2))`
`=sqrt(16+64+16)`
`=sqrt96=4sqrt6`