Home
Class 12
MATHS
Let overset(to)(a) =2hat(i) +hat(j) + h...

Let `overset(to)(a) =2hat(i) +hat(j) + hat(k), overset(to)(b) =hat(i) + 2hat(j) -hat(k)` and a unit vector `overset(to)(c )` be coplanar. If `overset(to)(c )` is perpendicular to `overset(to)(a) " then " overset(to)(c )` is equal to

A

`(1)/(sqrt(2) ) (-hat(j) +hat(k))`

B

`(1)/(sqrt(3)) (-hat(i) -hat(j) -hat(k))`

C

`(1)/(sqrt(5))(hat(i) -2hat(j))`

D

`(1)/(sqrt(5)) (hat(i) -hat(j)-hat(k))`

Text Solution

Verified by Experts

The correct Answer is:
A
It is given that `vec( c)` is coplanar with`vec(a) " and " vec(b)` we take
`vec(c )= P vec(a) + q vec(b)`
Where P and q are scalars.
Since `vec(C ) bot vec(a) rArr vec(c ) ". " vec(a) =0`
Taking dot product of `vec(a) .` in Eq (i) we get
`vec(C) ". " vec(a) =p vec(a) ". " vec(a) + q vec(b) "." vec(a) rArr 0= p|vec(a)|^(2) +q |vec(b)". " vec(a)|`
`[underset(=2+2-1=)underset(vec(a) ". " vec(b) =(2hat(i)+hat(j)+hat(k)).(vec(i)+2hat(j)-hat(k)))underset(|vec(a)|=sqrt(2^(2)+1+1)=sqrt(6).)( :' vec(a) =2hat(i) +hat(j) + hat(k))]`
`rArr 0 = p .6 + q .3 rArr q=-2p`
On putting in Eq . (i) we get
`vec(c ) =p vec(a) + vec(b) (-2p)`
`rArr vec(c ) =p [(2hat(i) + hat(j) + hat(k)) -2(hat(i) 2hat(j) - hat(k))]`
`rArr vec(c) = p^(2) (sqrt(18))^(2) rArr |vec(c )|^(2) =p^(2) . 18 `
`rArr 1=p^(2) .18`
`rArr p^(2) =(1)/(18) rArr p= +- (1)/(3sqrt(2))`
`:. vec(c ) =+- (1(-hat(j) +hat(k)))/(sqrt(2))`
Promotional Banner

Similar Questions

Explore conceptually related problems

Let overset(to)(a) = hat(i) - hat(j) , vecb=-hat(j) - hat(k) , overset(to)( c) =- hat(i) - hat(k) . If overset(to)(d) is a unit vector such that overset(to)(a).vec(d) =0= [ vec(b) vec(c ) vecd] then overset(to)(d) equals

If overset(to)(a) = (hat(i) + hat(j) + hat(k)) , overset(to)(a) .overset(to)(b) =1 " and " overset(to)(a) xx overset(to)(b) = hat(j) - hat(k), " then " overset(to)(b) is equal to

If overset(to)(a) =hat(i) - hat(k) , overset(to)(b) = x hat(i) + hat(j) + (1-x) hat(k) and vec(c ) =y hat(i) +x hat(j) + (1+x-y) hat(k) . "Then " [overset(to)(a) , overset(to)(b) , overset(to)( c) ] depends on

Let overset(to)(a) =a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k) , overset(to)(b) = b_(1) hat(i) +b_(2) hat(j) +b_(3) hat(k) " and " overset(to)(c) = c_(1) hat(i) +c_(2) hat(j) + c_(3) hat(k) be three non- zero vectors such that overset(to)(c ) is a unit vectors perpendicular to both the vectors overset(to)(c ) and overset(to)(b) . If the angle between overset(to)(a) " and " overset(to)(n) is (pi)/(6) then |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}|^2 is equal to

A vector perpendicular to both hat(i) + hat(j) + hat(k) and 2hat(i) + hat(j) + 3hat(k) is,

Let a= hat(i) - hat(j) , b=hat(i) + hat(j) + hat(k) and c be a vector such that axx c + b=0 " and " a. c =4, then |c|^(2) is equal to

Show that the vectors vec a = hat i - 2 hat j + 3 hat k, vec b = -2 hat i + 3 hat j - 4 hat k and vec c = hat i - 3 hat j + 5 hat k are coplanar.

Show that the vectors are coplanar 2hat(i)+3hat(j)+hat(k),hat(i)-hat(j),7hat(i)+3hat(j)+2 hat(k)

Show that the vectors vec a = hat i - 2 hat j + 3 hat k , vec b = -2 hat i +3hat j - 4 hat k and vec c = hat i - 3 hat j + 5 hat k are coplanar.