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Question
Let `vec"a", vec"b", vec"c"` be unit vectors such that `vec"a" * vec"b" = vec"a"*vec"c"` = 0 and the angle between `vec"b"` and `vec"c"` is `pi/3`. Prove that `vec"a" = +- 2/sqrt(3) (vec"b" xx vec"c")`
Solution
Given `vec"a", vec"b", vec"c"` are unit vectors.
∴ `|vec"a"|` = 1
`|vec"b"|` = 1
`|vec"c"|` = 1
Also `vec"a" * vec"b"` = 0, `vec"a" * vec"c"` = 0
Angle between `vec"b"` and `vec"c" = pi/3`
`vec"a" * vec"b"` = 0
⇒ `vec"a"` ⊥r `vec"b"`
`vec"a" * vec"c"` = 0
⇒ `vec"a"` ⊥r `vec"c"`
∴ `vec"a"` is perpendicular to both `vec"b"` and `vec"c"`
`vec"b" xx vec"c" = |vec"b"||vec"c"| sin pi/3 hat"n"`
When `hat"n"` is a unit vector perpendicular to both `vec"b"` and `vec"c"` which is `vec"a"`.
`vec"b" xx vec"c" = 1 xx 1 xx sqrt(3)/2 xx hat"n"`
`+- 2/sqrt(3) (vec"b" xx vec"c") = +- 2/sqrt(3) xx sqrt(3)/2 xx hat"n"`
`+- 2/sqrt(3) (vec"b" xx vec"c") = +- hat"n"` .......(1)
`+- hat"n"` is a unit vector perpendicular to both `vec"b"` and `vec"c"` which is `vec"a"`
(1) ⇒ `+- 2/sqrt(3) (vec"b" xx vec"c") = vec"a"`
`vec"a" = +- 2/sqrt(3) (vec"b" xx vec"c")`
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