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Question
Three vectors `vec"a", vec"b"` and `vec"c"` are such that `|vec"a"| = 2, |vec"b"| = 3, |vec"c"| = 4`, and `vec"a" + vec"b" + vec"c" = vec0`. Find `4vec"a"*vec"b" + 3vec"b"*vec"c" + 3vec"c"*vec"a"`
Solution
Given `|vec"a"| = 2, |vec"b"| = 3, |vec"c"| = 4`
Also `vec"a" + vec"b" + vec"c" = vec0`
`vec"a" + vec"b" = vec"c"`
`(vec"a" + vec"b")^2 = (- vec"c")^2`
`vec"a"^2 + vec"b"^2 + 2vec"a"*vec"b" = vec"c"^2`
`|vec"a"|^2 + |vec"b"|^2 + 2vec"a"*vec"b" = |vec"c"|^2`
`2^2 + 3^2 + 2vec"a"*vec"b" = 4^2`
`4 + 9 + 2vec"a"*vec"b"` = 16
`2vec"a"*vec"b"` = 16 – 13
`vec"a" * vec"b" = 3/2`
`vec"a" + vec"b" + vec"c" = vec0`
`(vec"a" + vec"b" + vec"c")^2 = (vec0)^2`
`vec"a"^2 + vec"b"^2 + vec"c"^2 + 2vec"a"*vec"b" + 2vec"b"*vec"c" + 2vec"c"*vec"a"` = 0
`|vec"a"|^2 + |vec"b"|^2 + |vec"c"|^2 + 2(vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a")` = 0
`2^2 + 3^2 + 4^2 + 2(vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a")` = 0
`4 + 9 + 16 + 2(vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a")` = 0
`2(vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a")` = – 29
`vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a" = - 29/2`
`3(vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a") = – 29/2 xx 3`
`vec"a"*vec"b" + vec"a"*vec"b" + 3vec"b"*vec"c" + 3vec"c"*vec"a" = -29/2 xx 3 + vec"a"*vec"b"`
`4vec"a"*vec"b" + vec"b"*vec"c" + 3vec"c"*vec"a" = - 29/2 xx 3 + 3/2`
= `3/2 (- 29 + 1)`
= `3/2 xx - 28`
`4vec"a"*vec"b" + 3vec"b"*vec"c" + 3vec"c"*vec"a"` = 3 × – 14
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