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The vector(s) which is/are coplanar wi...

The vector(s) which is/are coplanar with vectors ` hat i+ hat j+2 hat ka n d hat i+2 hat j+ hat k ,` and perpendicular to vector ` hat i+ hat j+ hat k ,` is/are

A

`hat(j) -hat(k)`

B

`-hat(i) + hat(j)`

C

`hat(i)-hat(j)`

D

`-hat(j)+hat(k)`

Text Solution

Verified by Experts

The correct Answer is:
AD
Let `vec(a) = hat(i) + hat(j) + 2hat(k) , vec(b) = hat(i) _+2hat(j) + hat(k)`
and `vec(c ) = hat(i) + hat(j) + hat(k)`
`:. ` A vector coplanar to `vec(a) " and " vec(b)` and perendicular to `vec (c )`
`= lambda (vec(a) xx vec(b)) xx vec(c ) = lambda {(vec(a) "." vec(c)) vec(v) - (vec(b) "." vec(c))vec(a)}`
`=lambda {(1+1 +4 )(hat(i) +2hat(j) +hat(k)) -(1+2 +1) (hat(i) + hat(j) +2hat(k))}`
`= lambda {(6 hat(i) + 12 hat(j) + 6 hat(k) -6hat(i) -6hat(j) - 12 hat(k)}`
`= lambda (6 hat(j) 6 hat(k)} = 6lambda {hat(j) - hat(k)}`
For `lambda = (1)/(6) rArr ` Option (a) is correct
and for `lambda =- (1)/(6) rArr` Option (d) is correct .
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