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")`

Answer 1

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")`