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If overset(to)(a),overset(to)(b),overset...

If `overset(to)(a),overset(to)(b),overset(to)(c )` are three non-zero , non-coplanar vectors and `overset(to)(b)_(1)=overset(to)(b) -.(overset(to)b.overset(to)(a))/(|overset(to)(a)|) overset(to)(a) , overset(to)(b)_(2) +.(overset(to)(b).overset(to)(a))/(|overset(to)(a)|^(2))overset(to)(a)`
`overset(to)(c)_(1) =overset(to)(c)-.(overset(to)(c).overset(to)a)/(|overset(to)(a)|^(2))overset(to)(a)-.(overset(to)(c).overset(to)(b))/(|overset(to)(b)|^(2))overset(to)(b),overset(to)c_(2)=overset(to)(c) -.(overset(to)(c).overset(to)(a))/(|overset(to)(a)|^(2))overset(to)(a)-.(overset(to)(c).overset(to)(b)_(1))/(|overset(to)(b)|^(2))overset(to)(b)_(1)`
`overset(to)(c)_(3) =overset(to)(c) -.(overset(to)(c).overset(to)(a))/(|overset(to)(a)|^(2))overset(to)(a)-.(overset(to)(c).overset(to)(b)_(2))/(|overset(to)(b)_(2)|^(2))overset(to)(b)_(2).overset(to)(c)_(4)=overset(to)(a) -.(overset(to)(c).overset(to)(a))/(|overset(to)(a)|^(2))overset(to)(a)`
Then which of the following is a set of mutually orthogonal vectors ?

A

`{overset(to)(a),overset(to)(b)_(1),overset(to)(c)_(1)}`

B

`{overset(to)(a),overset(to)(b)_(1),overset(to)(c)_(2)}`

C

`{overset(to)(a),overset(to)(b)_(2),overset(to)(c)_(3)}`

D

`{overset(to)(a),overset(to)(b)_(3),overset(to)(c)_(4)}`

Text Solution

Verified by Experts

The correct Answer is:
B
Since `vec(b)_(1) = vec(b) - (vec(b).vec(a))/(|vec(a)|^(2)) vec(a) , vec(b)_(1) =vec(b) + (vec(b).vec(a))/(|vec(a)|^(2)) vec(a)`
`" and " vec( c)_(1) =vec(c ) - (vec( c) "."vec( a))/(|vec(a)|^(2)) vec(a) - (vec(c )"."vec(b))/(|vec(b)|^(2)) vec(b) vec( c)_(2) = vec( c) - (vec( c). vec(a))/(|vec(a)|^(2)) vec(a) - (vec(c ) "."vec(b)_(1))/(|vec(b)|^(2)) vec(b)_(1)`
`vec(c)_(3) =vec(c) -(vec(c ).vec(a))/(|vec(a)|^(2))vec(a) -(vec(c ).vec(b)_(2))/(|vec(b)_(2)|^(2)) vec(b)_(2), vec(c )_(4) =vec(a) -(vec(c ). vec(a))/(|vec(a)|^(2))vec(a)`
Which shows `vec(a). vec(b)_(1) =0=vec(a) . vec(c )_(2)=vec(b)_(1).vec(c )_(2)`
So `{vec(a),vec(b)_(1),vec(c )_(2)}` are mutually orthogonal vectors.
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