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If \(\overrightarrow {a}\),\(\overrightarrow {b}\),\(\overrightarrow {c}\) are three vectors such that\( [\overrightarrow {a}+\overrightarrow {b}+\overrightarrow {c}=0] \)then prove that \(\overrightarrow {a}\times \overrightarrow {b}=\overrightarrow {b}\times \overrightarrow {c}=\overrightarrow {c} \times\overrightarrow {a}\)
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If \(\overrightarrow {a}\),\(\overrightarrow {b}\),\(\overrightarrow {c}\) are three vectors such that\( [\overrightarrow {a}+\overrightarrow {b}+\overrightarrow {c}=0] \)then prove that \(\overrightarrow {a}\times \overrightarrow {b}=\overrightarrow {b}\times \overrightarrow {c}=\overrightarrow {c} \times\overrightarrow {a}\)

Answer

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Solution

We have, \(\overrightarrow {a}+\overrightarrow {b}+\overrightarrow {c}=0.\)
\(\overrightarrow {a}\times (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow {c})=\overrightarrow {a}\times \overrightarrow {0}\)
\(\overrightarrow {a}\times \overrightarrow{a}+\overrightarrow {a}\times \overrightarrow {b}+\overrightarrow{a}\times \overrightarrow {c}=0\)
\(\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow {c}\times \overrightarrow {a}\)
Now,
\(\overrightarrow {b}\times (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})=\overrightarrow{a}\times \overrightarrow {0} \)
\(\overrightarrow{b}\times \overrightarrow{a}+\overrightarrow{b}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}=0\)
\(\overrightarrow{a}\times \overrightarrow {b}=\overrightarrow {b}\times \overrightarrow{c}\)
From above we have,
\(\overrightarrow {a}\times \overrightarrow {b}=\overrightarrow {b}\times \overrightarrow {c}=\overrightarrow {c} \times\overrightarrow {a}\)
Hence, proved.
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