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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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## QuestionMathsClass 10

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

See the solution below:
4.6
4.6

## 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.