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Suppose that vec p,vecqand vecr are thr...

Suppose that `vec p,vecqand vecr` are three non- coplaner in `R^(3)` ,Let the components of a vector`vecs` along `vecp , vec q and vecr` be 4,3, and 5, respectively , if the components this vector `vec s` along `(-vecp+vec q +vecr),(vecp-vecq+vecr) and (-vecp-vecq+vecr)` are x, y and z , respectively , then the value of `2x+y+z` is

A

`|overset(to)(a)|`

B

`|overset(to)(u)|+|overset(to)(u).overset(to)(a)|`

C

`|overset(to)(u)|+|overset(to)(u).overset(to)(a)|`

D

`|overset(to)(u)|+|overset(to)(u).(overset(to)(a)+overset(to)(b))`

Text Solution

Verified by Experts

The correct Answer is:
A, C
Let 0 be the angle between `vec(a) " and " vec(b) ` . Since `vec(a) " and " vec(b) ` are non- collinear vectors then 0 `ne ` 0 and `0 ne pi.`
we have `vec(a) ". " vec(b) =|vec(a)|vec(a)| " cos 0 "`
`= " cos 0 " " " [ :' |vec(a) |=1 , |vec(b)| =1 " given "]`
` " Now " vec(u) = vec(a) - (vec(a) ". " vec(b)) vec(b) rArr |vec(u) | = |vec(a) ". " vec(b) |`
`rArr |vec(u)|^(2) =|vec(a)|^(2)-(vec(a)". " vec(b)) vec(b)|^(2)`
`rArr |vec(u)|^(2) =1 + cos^(2) 0 rArr |vec(u)|^(2) - sin^(2) 0`
Also `vec(v) = vec(a) xx vec(b)`
`rArr |vec(v)|^(2)=|vec(a)xx vec(b)|^(2) rArr |vec(v)|^(2) = |a|^(2)|b|^(2) sin^(2) 0`
`rArr |vec(v)|^(2) = sin^(2) 0 :. |vec(u)|^(2) =|vec(v)|^(2)`
Now , `vec(u)"." vec(a) =[vec(a)- (vec(a)"." vec(b))vec(b)]"." vec(a) =vec(a)"." vec(a) - (vec(a)"."vec(b)) (vec(b)"."vec(a))`
`= (vec(a))^(2) - cos^(2) 0=1 - cos^(2) 0 = sin^(2) 0`
`:. |vec(u)|+|vec(u)"."vec(a)| =sin^(2) 0 ne |vec(v)|`
`vec(u)"." vec(b) =[vec(a) - (vec(a) ". " vec(b)) vec(b)] "." vec(b)`
`= vec(a) "." vec(b) -(vec(a) ". " vec(b)) (vec(b)". "vec(b)) =vec(a) ". "vec(b) - vec(a) ". " vec(b) |vec(b)|^(2)`
`= vec(a) ". " vec(b) -vec(a) ". " vec(b) =0`
Also `vec(u)". " (vec(a)+vec(b))=|vec(u)". " vec(a) + vec(u) ". " vec(b) = vec(u) ". " vec(a)`
`rArr |vec(u)|+ |vec(u) ". " (vec(a) + vec(b)) =|vec(u)| + vec(u) ". " vec(a) ne |vec(v)|`
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