Let ` vec a , vec b ,a n d vec c`are vectors such that `| vec a|=3,| vec b|=4a n d| vec c|=5,a n d( vec a+ vec b)`is perpendicular to ` vec c ,( vec b+ vec c)`is perpendicular to ` vec aa n d( vec c+ vec a)`is perpendicular to ` vec bdot`Then find the value of `| vec a+ vec b+ vec c|`.
Let `vec(a), vec(b), vec(c)` be the given vectors such that
`{|vec(a)|=3, |vec(b)|=4, |vec(c)|=5}`, …(i)
`{:(vec(a).(vec(b)+vec(c))=0),(vec(b).(vec(c)+vec(a))=0),(vec(c).(vec(a)+vec(b))=0):}}`. …(ii)
`:. |vec(a)+vec(b)+vec(c)|^(2)=(vec(a)+vec(b)+vec(c)).(vec(a)+vec(b)+vec(c))`
`=vec(a).vec(a)+vec(a).(vec(b)+vec(c))+vec(b).(vec(c)+vec(a))+vec(b).vec(b)+vec(c).(vec(a)+vec(b))+vec(c).vec(c)`
`=|vec(a)|^(2)+|vec(b)|^(2)+|vec(c)|^(2)` [using (ii)]
`=(3^(2)+4^(2)+5^(2))=(9+16+25)=50`
Hence, `|vec(a)+vec(b)+vec(c)|=sqrt(50)=5sqrt(2)`.