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"`

Answer 1

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`

Adding both sides b `vec"a"*vec"b"`\

`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

= – 42