Vectors & 3D Geometry
Non-coplanar selections from vertex vectors of cube
Grade Class 12
Question:
Consider the set of eight vectors $V=\{a\hat{i}+b\hat{j}+c\hat{k}: a,b,c\in\{-1,1\}\}$. The number of ways three non-coplanar vectors can be chosen from $V$ equals
Step-by-Step Solution
Key Concept: All 8 vectors form antipodal pairs. Three vectors are coplanar iff two are antipodal (collinear through origin). Non-coplanar = total $\binom{8}{3}$ minus coplanar selections.
32 non-coplanar selections.
Correct Answer: 32