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A Unit Vector Perpendicular to Both ^ I + ^ J and ^ J + ^ K is - Mathematics

MCQ

A unit vector perpendicular to both \[\hat{ i }  + \hat{ j } \text{ and }  \hat{ j } + \hat{ k } \] is

 

Options

  • \[\hat{ i }  - \hat{ j }  + \hat{ k } \]

  • \[\hat{ i }  + \hat{ j }  + \hat{ k } \] 

  • \[ \frac1 {\sqrt3}  ( \hat{ i }  + \hat{ j }  + \hat{ k } ) \] 

  • \[ \frac1 {\sqrt3}  ( \hat{ i }  - \hat{ j }  + \hat{ k } ) \] 

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Solution

\[ \frac1 {\sqrt3}  ( \hat{ i }  - \hat{ j }  + \hat{ k } ) \] 

\[\text{ Let } :\]

\[ \vec{a} = \hat{ i }  + \hat{ j }  + 0 \hat{ k }  \]

\[ \vec{b} = 0 \hat{ i } + \hat{ j }  + \hat{ k }  \]

\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i }  & \hat{ j }  & \hat{ k }  \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{vmatrix}\]

\[ = \hat{ i  } - \hat{ j }  + \hat{ k } \]

\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{1 + 1 + 1}\]

\[ = \sqrt{3}\]

\[\text{ Unit vector perpendicular to } \vec{a} \text{ and } \vec{b} =\frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|} = \frac{\hat{ i }  - \hat{ j }  + \hat{ k } }{\sqrt{3}}\]

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APPEARS IN

RD Sharma Class 12 Maths
Chapter 25 Vector or Cross Product
MCQ | Q 8 | Page 35
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