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find 3- dimensional vectors overset(t...

find 3- dimensional vectors `overset(to)(v)_(1) , overset(to)(v)_(2), overset(to)(v)_(3)` satisfying
`overset(to)(v)_(1),overset(to)(v)_(1) =4, overset(to)(v)_(1).overset(to)(v)_(2)=-2overset(to)(v)_(1).overset(to)(v)_(3)-6`
`overset(to)(v)_(2).overset(to)(v)_(2)=2,overset(to)(v)_(2).overset(to)(v)_(3) =-5 , overset(to)(v)_(3).overset(to)(v)_(3)=29`

Text Solution

Verified by Experts

The correct Answer is:
`overset(to)(v)_(1) = 2hat(i) overset(to)(v)_(2) =- hat(i) + hat(j) " and " overset(to)(v)_(3) = 3hat(i) +2hat(j) +-4 hat(k)`
We have `|vec(to)_(1)| =2,|vec(v)_(2)|=sqrt(2) " and " |vec(V)_(3)| = sqrt(39)` If 0 is the angle between `vec(v)_(1)" and " vec(v)_(2)` then
`2sqrt(2) " cos " 0 = - 2`
`rArr " cos " 0=- (1)/(sqrt(2))`
`rArr 0= 135^(@)` Since any two vectors are always coplanar and data is not sufficient so we can assume `vec(v)_(1) " and " vec(v)_(2)` in x-y plane.
`vec(v)_(1) = 2hat(i)`
`vec(v)_(2) =- hat(i) + hat(j)`

`" and " vec(v)_(3) =alpha hat(i) + beta hat(j) + gamma hat(k)`
`" Since " vec(v)_(3) - vec(v)_(1) =6 = 2alpha rArr alpha =3`
Also ` vec(v)_(3) "."vec(v)_(2) =-5 = alpha +- beta rArr beta = +- 2`
`" and " vec(v)"."vec(3) = 29 = alpha^(2) + beta^(2) + gamma^(2) rArr gamma = +- 4`
Hence ` vec(v)_(3) = 3hat(i) +- 2hat(j) +- 4hat(k)`
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