Consider the set of eight vectors V-ai-bj-ck-a-b-c -1-1- Three non - coplanar vectors can be chosen from V in 2p ways- Then p is

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Standard XII

Mathematics

Combination Based on Geometry

Consider the ...

Question

Consider the set of eight vectors
V={ai+bj+ck}:a,b,c $ϵ {- 1, 1}$ . Three non - coplanar vectors can be chosen from V in $2^{p}$ ways, Then $p$ is ___

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Solution

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)
These are 4 diagonals of a cube and their opposites. For non coplanar vectors first we select 3 groups of diagonals and its opposite in $^{4} C_{3}$ ways. Then one vector from each group can be selected in $2 \times 2 \times 2 w a y s .$
$∴ T o t a l w a y s = 4 C_{3} \times 2 \times 2 \times 2 = 32 = 2^{5} ∴ p = 5$

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Consider the set of eight vectors V={ai+bj+ck}:a,b,c ϵ{−1,1}. Three non - coplanar vectors can be chosen from V in 2p ways, Then p is ___

Consider the set of eight vectors
V={ai+bj+ck}:a,b,c $ϵ {- 1, 1}$ . Three non - coplanar vectors can be chosen from V in $2^{p}$ ways, Then $p$ is ___