# 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

• $\hat{ i } - \hat{ j } + \hat{ k }$

• $\hat{ i } + \hat{ j } + \hat{ k }$

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

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

#### 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