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Let   \vec a , \vec b , \vec c are three non coplanar vectors , such that [\vec a \: \: \vec b\: \: \vec c ]= 2  then reciprocal system of vectors will form a parallelopiped with volume 

Option: 1

1/2


Option: 2

1


Option: 3

3/2


Option: 4

2


Answers (1)

best_answer

As we have learned

Reciprocal System of Vectors -

\vec{a}\ {}'=\frac{\vec{b}\times \vec{c}}{\left [\vec{a} \ \vec{b} \ \vec{c}\right ]}

\vec{b}\ {}'=\frac{\vec{c}\times \vec{a}}{\left [\vec{a} \ \vec{b} \ \vec{c}\right ]}

\vec{c}\ {}'=\frac{\vec{a}\times \vec{b}}{\left [\vec{a} \ \vec{b} \ \vec{c}\right ]}

 

- wherein

\vec{a}, \vec{b}, \vec{c}are three vectors.

 

 Reciprocal system of vectors will be \frac{\vec b \times \vec c }{[\vec a\: \: \vec b \: \: \vec c]},\frac{\vec c \times \vec a }{[\vec a\: \: \vec b \: \: \vec c]},\frac{\vec a \times \vec b }{[\vec a\: \: \vec b \: \: \vec c]},

i.e . \frac{\vec b\times \vec c}{2}, \frac{\vec c \times \vec a }{2}, \frac{\vec a \times \vec b }{2}

 volume of paralleopiped = \left |\left [ \frac{\vec b\times \vec c}{2}\: \: \frac{\vec c \times \vec a }{2}\: \: \frac{\vec a \times \vec b }{2} \right ] \right |

\left | \frac{1}{8}\left [ {\vec b\times \vec c}\: \: \: \: {\vec c \times \vec a }\: \:\: \: {\vec a \times \vec b } \right ] \right |= 1/8 \left [ \vec a \: \: \vec b \: \: \vec c \right ]^2

1/8\times 4 = 1/2

 

 

 

 

 

Posted by

Suraj Bhandari

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