Let `vecA , vecB and vecC` be vectors of legth , 3,4and 5 respectively. Let `vecA` be perpendicular to `vecB + vecC, vecB " to " vecC + vecA and vecC " to" vecA + vecB` then the length of vector `vecA + vecB+ vecC` is __________.

Answer

`5sqrt2`

Explanation

Given that `|vecA|=3, |vecB|=4, |vecC|=5`
`vecAbot (vecB + vecC) Rightarrow vecA. (vecB +vecC) =0`

`Rightarrow vecA.vecB + vecA.vecC=0`
` vecB bot (vecC +vecA)RightarrowvecB.(vecC+vecA_=0`
`Rightarrow vecB.vecC+vecB.vecA=0`
`vecCbot (vecA+vecB) RightarrowvecC. (vecA+vecB)=0`
` Rightarrow vecC.vecA+vecC.vecdB=0`
Adding (i), (ii) and (iii) we get
`2(vecA.vecBr+vecB.vecC+vecC.vecA)=0`
Now , `|vecA + vecB + vecC|^(2)`
`(vecA + vecB+vecC).(vecA + vecB+vecC)`
`|vecA|^(2)+|vecB|^(2)+|vecC|^(2)`
`+2(vecA.vecB + vecB.vecC+vecC.vecA)`
9+16+25+0
= 50
`|vecA + vecB +vecC|= 5sqrt2`