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Question
If `|vec"a"|= 5, |vec"b"| = 6, |vec"c"| = 7` and `vec"a" + vec"b" + vec"c" = vec"0"`, find `vec"a" * vec"b" + vec"b" *vec"c" + vec"c" * vec"a"`
Solution
Given `|vec"a"|= 5, |vec"b"| = 6, |vec"c"| = 7`
`vec"a" + vec"b" + vec"c" = vec"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"0"^2 = |vec"a"|^2 + |vec"b"|^2 + |vec"c"|^2 + 2(vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a")`
0 = `5^2 + 6^2 + 7^2 + 2(vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a")`
0 = `25 + 36 + 49 + 2(vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a")`
`2(vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a") = - 110`
`vec"a" * vec"b" + vec"b" *vec"c" + vec"c" * vec"a" = (-110)/2 = - 55`
`vec"a" * vec"b" + vec"b" *vec"c" + vec"c" * vec"a" = - 55`
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