Consider the set of eight vectors V = \ a ^ i + b ^ j + c ^ k : a, b, c ∈ \ - 1 , 1\ \ . Three non-coplanar vectors can be chosen fro m V in 2 ^ p ways. Then p is

JEE Advanced 2013Paper 1

JEE Advanced 2013 · Paper 1 · Mathematics · Vector Algebra

Consider the set of eight vectors V = \ a ^ i + b ^ j + c ^ k : a,...

Consider the set of eight vectors

\text{V} = \left\{\text{a} \overset{^}{i} + \text{b} \overset{^}{\text{j}} + \text{c} \overset{^}{\text{k}} : a, b, c \in \left\{- 1 , 1\right\}\right\}. Three non-coplanar vectors can be chosen

fro

m \text{V} in

{2}^

{\text{p}} ways. Then \text{p} is

\overset{\to}{\text{x}} &equiv; \overset{^}{\text{i}} + \overset{^}{\text{j}} + \overset{^}{\text{k}}

\overset{\to}{\text{y}} &equiv; \overset{^}{\text{i}} + \overset{^}{\text{j}} - \overset{^}{\text{k}}

\overset{\to}{\text{z}} &equiv; \overset{^}{\text{i}} + \overset{^}{\text{j}} + \overset{^}{\text{k}}

\overset{\to}{&omega;} = \overset{^}{\text{i}} - \overset{^}{\text{j}} - \overset{^}{\text{k}}

{\overset{&macr;}{\text{x}}}^{&prime;} = - \overset{^}{\text{i}} - \overset{^}{\text{j}} - \overset{^}{\text{k}}

{\overset{\to}{\text{y}}}^{&prime;} = - \overset{^}{\text{i}} - \overset{^}{\text{j}} + \overset{^}{\text{k}}

{\overset{\to}{\text{z}}}^{&prime;} = - \overset{^}{\text{i}} + \overset{^}{\text{j}} - \overset{^}{\text{k}}

{\overset{\to}{&omega;}}^{&prime;} = - \overset{^}{\text{i}} + \overset{^}{\text{j}} + \overset{^}{\text{k}}

If we take

\overset{\to}{\text{x}} \text{,} {\overset{\to}{\text{x}}}^{&prime;}

and any one of remaining six, vectors will always be coplanar

&there4;

No. of coplanar vectors = 6
Similarly on taking

\overset{\to}{\text{y}} \text{,} {\overset{\to}{\text{y}}}^{&prime;} = 6

\text{z} \text{,} {\overset{\to}{\text{z}}}^{&prime;} = 6

&omega; \text{,} {\overset{\to}{&omega;}}^{&prime;} = 6

&there4;

No. of set coplanar vectors = 24

{}^{8} {\text{C}}_{3} - 2 4 = 3 2