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

Given 8 vectors are (1,1,1),(-1,1,1),(1,-1,1),(1,1,-1),(-1,-1,1),(1,-1,-1),(-1,1,-1),(-1,-1,-1) there are 4 diagonals of a cube. Now, for 3 non-coplanar vectors first we select 3 groups of diagonals and its opposite in `.^(4)C_(3)=4` ways. Then one vector from each group cann be selected in `2xx2xx2=8` ways.
`therefore`Total ways`=4xx8=32=2^(5)=2^(p)` (given)
Hence, `p=5`