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Mathematics Question on Vector Algebra

The scalar A→.[(B→+C→)×(A→+B→+C→)]\overrightarrow{A}.[(\overrightarrow{B}+\overrightarrow{C})\times(\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C})] eqy=ual

A

0

B

[A→.B→C→]+[B→.C→A→][\overrightarrow{A}.\overrightarrow{B}\overrightarrow{C}]+[\overrightarrow{B}.\overrightarrow{C}\overrightarrow{A}]

C

[A→.B→C→][\overrightarrow{A}.\overrightarrow{B}\overrightarrow{C}]

D

None of these

Answer

0

Explanation

Solution

\overrightarrow{A}.\\{(\overrightarrow{B}+\overrightarrow{B})\times(\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C})
[∵it is a scalar triple product of three vectors of the form A→,B→+C→,A→+B→+C→][\because\text{it is a scalar triple product of three vectors of the form } \overrightarrow{A},\overrightarrow{B}+\overrightarrow{C},\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}]
=A→(B→×A→+B→×B→+B→×C→+C→×A→+C→×B→+C→×C→)=\overrightarrow{A}(\overrightarrow{B}\times\overrightarrow{A}+\overrightarrow{B}\times\overrightarrow{B}+\overrightarrow{B}\times\overrightarrow{C}+\overrightarrow{C}\times\overrightarrow{A}+\overrightarrow{C}\times\overrightarrow{B}+\overrightarrow{C}\times\overrightarrow{C})
=A→.(B→×A→+A→.(B→×C→)+A→.(C→×B→)=\overrightarrow{A}.(\overrightarrow{B}\times\overrightarrow{A}+\overrightarrow{A}.(\overrightarrow{B}\times\overrightarrow{C})+\overrightarrow{A}.(\overrightarrow{C}\times\overrightarrow{B})
[A→B→A→]−[A→B→C→]=0[\overrightarrow{A}\overrightarrow{B}\overrightarrow{A}]-[\overrightarrow{A}\overrightarrow{B}\overrightarrow{C}]=0