Question

If \[\vec{c}\] s perpendicular to both \[\vec{a} \text{ and } \vec{b}\] then prove that it is perpendicular to both \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b}\] 

Answer 1

\[\text{ Given that } \vec{c} \text{ is perpendicular to both } \vec{a} \text{ and } \vec{b} . \]
\[ \Rightarrow \vec{c} . \vec{a} = 0 \text{ and } \vec{c} . \vec{b} = 0 . . . \left( 1 \right)\]
\[\text{ Now },\]
\[ \vec{c} . \left( \vec{a} + \vec{b} \right) = \vec{c} . \vec{a} + \vec{c} . \vec{b} = 0 + 0 = 0................. \left[ \text{ From }\left( 1 \right) \right]\]
\[\text{ So }, \vec{c} \text{ is perpendicular to } \vec{a} + \vec{b} . \]
\[\text{ Again },\]
\[ \vec{c} . \left( \vec{a} - \vec{b} \right) = \vec{c} . \vec{a} - \vec{c} . \vec{b} = 0 - 0 = 0 ..................\left[ \text{ From }\left( 1 \right) \right]\]
\[\text{ So }, \vec{c} \text{ is perpendicular to } \vec{a} - \vec{b} . \]