Question

If a, b and c are three non-zero vectors such that each one of them being perpendicular to the sum of the other two vectors, then the value of `|a+b+c|^(2)` is
A. `|a|^(2)+|b|^(2)+|c|^(2)`
B. `|a|+|b|+|c|`
C. `2 (|a|^(2)+|b|^(2)+|c|^(2))`
D. `1/2 (|a|^(2)+|b|^(2)+|c|^(2))`

Answer 1

Correct Answer - A
According to the given condition, each vectors is perpendicular to the sum of the vectors.
`:. a.(b+c) =0`,
`b.(a+c) =0`
and `c.(a+b) = 0`
`rArr a.b+a.c = 0, b.a + b.c = 0`
and `c.a + c.b = 0`
`rArr 2(a.b+b.c +c.a) = 0`
`rArr a.b+b.c+c.a= 0"...."(i)`
Now, `|a+b+c|^(2) = |a|^(2) + |b|^(2) + |c|^(2) + 2(a.b+b.c+c.a)`
`= |a|^(2) +|b|^(2) +|c|^(2) + 2(0)` [from eq. (i)]
`= |a|^(2) +|b|^(2) +|c|^(2)`