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Question
Let `vec"a", vec"b", vec"c"` be three vectors such that `|vec"a"| = 3, |vec"b"| = 4, |vec"c"| = 5` and each one of them being perpendicular to the sum of the other two, find `|vec"a" + vec"b" + vec"c"|`
Solution
Given `|vec"a"| = 3, |vec"b"| = 4, |vec"c"| = 5`
Also `vec"a" * (vec"b" + vec"c")` = 0,
`vec"b" * (vec"c" + vec"a")` = 0
`vec"c" * (vec"a" + vec"b")` = 0
`vec"a" * (vec"b" + vec"c")` = 0
`vec"a" * vec"b" + vec"a" * vec"c"` = 0 ........(1)
`vec"b" * (vec"c" + vec"a")` = 0
`vec"b" * vec"c" + vec"b" * vec"a"` = 0 ........(2)
`vec"c" * (vec"a" + vec"b")` = 0
`vec"c" * vec"a" + vec"c" * vec"b"` = 0 ........(3)
Equation (1) + (2) + (3) ⇒
`vec"a" * vec"b" + vec"a" * vec"c" + vec"b" * vec"c" + vec"b" * vec"a" + vec"c" * vec"a" + vec"c" * vec"b"` = 0
`vec"a" * vec"b" + vec"c" * vec"a" + vec"b" * vec"c" + vec"a" * vec"b" + vec"c" * vec"a" + vec"b" * vec"c"` = 0
`2vec"a" * vec"b" + 2vec"b" * 2vec"c" * vec"a"` = 0
`2(vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a")` = 0
`(vec"a" + vec"b" + vec"c")^2 = vec"a"^2 + vec"b"^2 + vec"c"^2 + 2vec"a"*vec"b" + 2vec"b"*vec"c" + 2vec"c"*vec"a"`
`|vec"a" + vec"b" + vec"c"|^2 = |vec"a"|^2 + |vec"b"|^2 + |vec"c"|^2 + 2(vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a")`
= `3^2 + 4^2 + 5^2 + 0`
= 9 + 16 + 25
= 25 + 25
`|vec"a" + vec"b" + vec"c"|^2` = 50
`|vec"a" + vec"b" + vec"c"| = sqrt(2 xx 25) = sqrt(50)`
`|vec"a" + vec"b" + vec"c"| = 5sqrt(2)`
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