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7  Show that the vectors \vec a , \vec b , \vec c  coplanar if \vec a + \vec b , \vec b + \vec c , \vec c + \vec a are coplanar

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vectors \vec a , \vec b , \vec c will be  coplanar if 

\left [ \vec a,\vec b, \vec c \right ]=0

now, it is given that

 \left [ \vec a+\vec b,\vec b+\vec c, \vec c+\vec a \right ]=0

( \vec a+\vec b)\cdot[(\vec b+\vec c)\times(\vec c+\vec a)] =0

( \vec a\cdot[(\vec b+\vec c)\times(\vec c+\vec a)]+( \vec b\cdot[(\vec b+\vec c)\times(\vec c+\vec a)] =0

now as we know that whenever two vectors are same in a vector triple product its value becomes zero,  so

\left [ \vec a+\vec b, \vec b+\vec c, \vec c+\vec a \right ]=\left [ \vec a, \vec b, \vec c \right ]+0+0+0+0+\left [ \vec a, \vec b, \vec c \right ]

\left [ \vec a+\vec b, \vec b+\vec c, \vec c+\vec a \right ]=2\left [ \vec a, \vec b, \vec c \right ]

Hence when  \vec a + \vec b , \vec b + \vec c , \vec c + \vec a is zero \left [ \vec a,\vec b, \vec c \right ]=0

Posted by

Pankaj Sanodiya

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