Consider the set of eight vectors V = {\({a\hat{i}}+b\hat{j}+c\hat{k}\

1.	Consider the set of eight vectors V = {\({a\hat{i}}+b\hat{j}+c\hat{k}\,;a,b,c ∈\){-1,1}}.Three non-coplanar vectors can be chosen from V in 2p ways. Then p is ____
Answer» Correct answer (5) Let (1, 1, 1), (-1, 1, 1), (1, -1, 1), (-1, -1, 1) be vectors \(\vec{a},\vec{b},\vec{c},\vec{d}\) rest of the vectors are \(-\vec{a},-\vec{b},-\vec{c},-\vec{d}\) and let us find the number of ways of selecting co–planar vectors. Observe that out of any 3 coplanar vectors 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 co–planar selections = ⁸C₃ - 24 = 32 = 2⁵ , p = 5

Consider the set of eight vectors V = {\({a\hat{i}}+b\hat{j}+c\hat{k}\,;a,b,c ∈\){-1,1}}.Three non-coplanar vectors can be chosen from V in 2p ways. Then p is ____

Answer»

Correct answer (5)

Let (1, 1, 1), (-1, 1, 1), (1, -1, 1), (-1, -1, 1) be vectors

\(\vec{a},\vec{b},\vec{c},\vec{d}\) rest of the vectors are \(-\vec{a},-\vec{b},-\vec{c},-\vec{d}\)

and let us find the number of ways of selecting co–planar vectors.

Observe that out of any 3 coplanar vectors 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 co–planar selections = ⁸C₃ - 24 = 32 = 2⁵ , p = 5

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