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Question
If `vec"a", vec"b", vec"c"` are three vectors such that `vec"a" + 2vec"b" + vec"c"` = 0 and `|vec"a"| = 3, |vec"b"| = 4, |vec"c"| = 7`, find the angle between `vec"a"` and `vec"b"`
Solution
`vec"a" + 2vec"b" + vec"c"` = 0
`|vec"a"| = 3, |vec"b"| = 4, |vec"c"| = 7`
`vec"a" + 2vec"b" = -vec"c"`
Squaring on both sides,
`(vec"a" + 2vec"b")^2 = (- vec"c")^2`
`vec"a"^2 + 4vec"b"^2 + 2vec"a" * 2vec"b" = vec"c"^2`
`|vec"a"|^2 + 4|vec"b"|^2 + 4vec"a" * vec"b" = |vec"c"|^2`
`3^2 + 4 xx 4^2 + 4 |vec"a"||vec"b"| cos theta = 7^2`
`9 + 64 + 4 xx 3 xx 4 cos theta` = 49
48 cos θ = 49 – 73
48 cos θ = – 24
cos θ = `- 24/48`
cos θ = `- 1/2`
cos θ = = `- cos pi/3`
cos θ = `cos(pi - pi/3)`
= `cos((3pi- pi)/3)`
cos θ = `cos ((2pi)/3)`
θ = `(2pi)/3`
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