If a vector a -is perpendicular to two non-collinear vectors b -and c - then show that a -is perpendicular to every vector in the plane of b -and c -

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Byju's Answer

Standard XII

Mathematics

Vector Triple Product

If a vector a...

Question

If a vector $\vec{a}$ is perpendicular to two non-collinear vectors $\vec{b} and \vec{c}, then show that \vec{a}$ is perpendicular to every vector in the plane of $\vec{b} and \vec{c} .$

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Solution

$Given that \vec{a} is perpendicular to \vec{b} and \vec{c} . \Rightarrow \vec{a} . \vec{b} = 0 and \vec{a} . \vec{c} =0 ... (1) Now, let \vec{r} be any vector in the plane of \vec{b} and \vec{c} . Then, \vec{r} is the linear combination of \vec{b} and \vec{c} . \vec{r} = x \vec{b} + y \vec{c}, for some x and y . Now, \vec{a} . \vec{r} = \vec{a} . (x \vec{b} + y \vec{c}) = x (\vec{a} . \vec{b}) + y (\vec{a} . \vec{c}) = x (0) + y (0) [From (1)] = 0 Thus, \vec{a} is perpendicular to \vec{r} . That is, \vec{a} is perpendicular to every vector in the plane of \vec{b} and \vec{c} .$

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If a vector a →is perpendicular to two non-collinear vectors b →and c →, then show that a →is perpendicular to every vector in the plane of b →and c →.

If a vector $\vec{a}$ is perpendicular to two non-collinear vectors $\vec{b} and \vec{c}, then show that \vec{a}$ is perpendicular to every vector in the plane of $\vec{b} and \vec{c} .$