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Question
If \[\vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = 0\] for some non-zero vector \[\vec{r} ,\] then the value of \[\left[ \vec{a} \vec{b} \vec{c} \right],\] is
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Solution
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\[\text { If } \vec{r} . \vec{a} = 0 \text { for some non - zero vector } \vec{r} ,\text { then either } \vec{a}\text { is a zero - vector or it is perpendicular to }\vec{r} . \]
\[\text { If one of } \vec{a} , \vec{b} , \bar{c} \text { is zero, then } \left[ \vec{a} \vec{b} \vec{c} \right] = 0 \]
\[\text { If all } \vec{a} , \vec{b} \text { and } \vec{c} \text { are non - zero, then they must be coplanar as they are perpendicular to vector } \vec{r} . \]
\[ \therefore \left[ \vec{a} \vec{b} \vec{c} \right] = 0\]
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