If a and b are two non-zero, non-collinear vectors, then `2[[a, b ,hati]]hati+2[[a,b,hatj]]hatj+2[[a,b,hatk]]hatk` is equal to

Options

A. `2(axxb)`
B. `axxb`
C. `a+b`
D. None of these

Answer

A

Explanation

Let `a=a_(1)hat(i)+a_(2)hat(j)+a_(3)hat(k) and b=b_(1)hat(i)+b_(2)hat(j)+b_(3)hat(k)`
Now , `[abhat(i)]=|{:(a_1,a_2,a_3),(b_1,b_2,b_3),(1,0,0):}|`
`=a_(1)(0-0)-a_2(0-b_3)+a_(3)(0-b_2)=a_2b_3-a_3b_2`
`therefore 2[ab hat(i)]hat(i)=2(a_2b_3-a_3b_2)hat(i)`
Similarly, `2[a b hat(j)]hat(j)=2(a_1b_2-a_1b_3)hat(j)`
and `2[a bhat(k)]hat(k)=2(a_2b_2-a_2b_1)hat(k)`
`therefore 2[ab hat(i)]hat(i)+2[abhat(j)]hat(j)+2[abhat(k)]hat(k)+[aba]`
`=2[(a_2b_3-a_(3)b_2)hat(i)+(a_3b_1-a_1b_3)hat(j)+0+(a_1b_2-a_2b_1)hat(k)]`
`=2(axxb)` .