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Question
If a vector \[\vec{a}\] is perpendicular to two non-collinear vectors \[\vec{b} \text{ and } \vec{c} , \text{ then show that } \vec{a}\] is perpendicular to every vector in the plane of \[\vec{b} \text{ and } \vec{c} .\]
Solution
\[\text{ Given that } \vec{a} \text{ is perpendicular to } \vec{b} \text{ and } \vec{c} .\]
\[ \Rightarrow \vec{a} . \vec{b} = 0 \text{ and } \vec{a} . \vec{c} =0 ... (1)\]
\[\text{ Now, let } \vec{r} \text{ be any vector in the plane of } \vec{b} \text{ and } \vec{c} .\]
\[\text{ Then }, \vec{r} \text{ is the linear combination of } \vec{b} \text{ and } \vec{c} .\]
\[ \vec{r} = \text{ x } \vec{b} + \text{ y }\vec{c} , \text{ for some x and y }.\]
\[\text{ Now },\]
\[ \vec{a} . \vec{r} \]
\[ = \vec{a} . \left( \text{ x } \vec{b} + \text{ y }\vec{c} \right)\]
\[ = x \left( \vec{a} . \vec{b} \right) + y \left( \vec{a} . \vec{c} \right)\]
\[ = x\left( 0 \right) + y\left( 0 \right) ..................[\text{ From } (1)]\]
\[ = 0\]
\[\text{ Thus }, \vec{a} \text{ is perpendicular to } \vec{r} .\]
\[\text{ That is, }\vec{a} \text{ is perpendicular to every vector in the plane of } \vec{b} \text{ and } \vec{c} .\]
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