Consider the set of eight vector `V={a hat i+b hat j+c hat k ; a ,bc in {-1,1}}dot` Three non-coplanar vectors can be chosen from `V` is `2^p` ways. Then `p` is_______.

Answer

5

Explanation

Let `(1, 1, 1), (-1, 1, 1),(1, -1, 1),(-1, -1, 1)` be vectors `veca, vecb, vecc, vecd`, rest of the vectors are `-veca, -vecb, -vecc, -vecd ` and let us find the number of ways of selecting coplanar vectors.
Observe that out of any three coplanar vecors two will be collinear (anti parallel).
Number of ways of selecting the anti-parallel pair =4
Number of ways of selecting the third vector =6
Total = 24
Number of non-coplanar selections
`" " = ""^(8)C_(3)- 24 = 32 = 2^(5), p=5`
`therefore p =5`