Let `a= 2hat(i)+ hat(j) -2hat(k) , b=hat(i) +hat(j) ` and c be a vectors such that `|c-a| =3, |(axxb)xx c|=3` and the angle between c and `axx b" is "30^(@)` . Then a. c is equal to

Options

A. `(25)/(8)`
B. `2`
C. `5`
D. `(1)/(8)`

Answer

B

Explanation

we have `a= 2hat(i)+ hat(j) -2hat(k)`
`rArr |a|= sqrt(4+1+4)=3`
`" and " b=hat (i) + hat(j)`
`rArr |b| = sqrt(1 +1)= sqrt(2)`
Now `|c-a|=3 rArr |c-a|^(2) =9`
` rArr (c-a) .(c-a) =9`
`rArr |c|^(2) +|a|^(2)- 2 c.a =9`
Again , `|(a xx b) xx c|=3`
`rArr | a xx b| | c| " sin " 30^(@) = 3`
`|c| = (6)/(|axxB|)`
But `axx b = |{:(hat(i),,hat(j),,hat(k)),(2,,1,,-2),(1,,1,,0):}| = 2hat(i) - 2hat(j) + hat(k)` ltbgt `:. |c| = (6)/(sqrt(4+4+1)) = 2`
From Eqs . (i) and (ii) we get
`(2)^(2) +(3)^(2) - 2c. a=9`
`rArr 4+9 -3c. a =9`
`rArr c.a =2`