Let `vec(A),vec(B),vec(C )` be vectors of length 3, 4, 5, respectively Let `overset(to)(A)` be perpendicular to `overset(to)(B) +overset(to)(C ) , overset(to)(B) " to " overset(to)( C) + overset(to)(A) " and " overset(to)(C )` to `overset(to)(A) +overset(to)(B)` then the length of vector `overset(to)(A) +overset(to)(B)+overset(to)(C )` is .......

Answer

`(5sqrt(2))`

Explanation

Given `|vec(A)| =3, |vec(B)| =4, |vec(C )|=5`
Since `vec(A)"."(vec(B)+vec(C )) =vec(B). (vec(C ) +vec(A)) = vec(C ) . (vec(A) +vec(B)) =0`
`:. |vec(A) +vec(B) + vec(C )|^(2) =|vec(A)|+|vec(B)|^(2)+|vec(C )|^(2)`
`+2(vec(A)"."vec(B)+vec(B)"."vec(C ) +vec(C ) "."vec(A))`
`=9 +16 +25 +0`
[from Eq. (i) , `vec(A)"."vec(B) +vec(B)"."vec(C) +vec(C )"."vec(A)=0]`
`:. |vec(A) + vec(B) + vec(C )|^(2)=50`
`rArr |vec(A)+vec(B)+vec(C )|=5 sqrt(2)`