Question

if a,b and c are three vectors such that |a|=3,|b|=4,|c|=5 and each one of them is perpendicular to the sum of the other two,then find |a+b+c|

Answer 1

a^ , b^ , c^ are three vector .

a/c to question ,

a^ perpendicular upon ( b^ +c^)

b^ perpendicular upon (c^ + a^)

c^ perpendicular upon (a^ + b^)

we know,
dot product of two perpendicular vector =0
so,

a^.(b^ + c^) = 0

b^( c^ +a^) = 0

c^(a^ + b^) =0

add all of this expression ,

a^.b^ + b^.c^ + c^.a^ =0

b^.( a^ + c^) = -c^.a^

but b^.(a^ + c^) =0

so, -c^.a^=0

hence, c^ perpendicular to a^

in the same way ,
a^ perpendicular to b^ and

c^ perpendicular to a^

| a + b + c | =√(a + b + c).(a + b + c )

=√(a + b + c )^2

=√(|a|^2 +|b|^2 +|c|^2+2a.b+2b.c+2c.a)

=√( |a|^2 + |b|^2 + |c|^2)

=√(3^2 + 4^2 + 5^2)
=5√2