Find a Unit Vector Perpendicular to Both the Vectors 4 ^ I − ^ J + 3 ^ K and − 2 ^ I + ^ J − 2 ^ K . - Mathematics

Sum

Find a unit vector perpendicular to both the vectors  $4 \hat{ i } - \hat{ j } + 3 \hat{ k } \text{ and } - 2 \hat{ i } + \hat{ j } - 2 \hat{ k } .$

Solution

$\text{ Given } :$
$\vec{a} = 4 \hat { i } - \hat{ j } + 3 \hat{ k }$
$\vec{b} = - 2 \hat{ i } + \hat{ j } - 2 \hat{ k }$
$\therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k} \\ 4 & - 1 & 3 \\ - 2 & 1 & - 2\end{vmatrix}$
$= \left( 2 - 3 \right) \hat{ i } - \left( - 8 + 6 \right) \hat{ j } + \left( 4 - 2 \right) \hat{ k }$
$= - \hat{ i } + 2 \hat{ j } + 2 \hat{ k }$
$\Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{1 + 2^2 + 2^2}$
$= \sqrt{9}$
$= 3$
$\text{ Unit vector perpendicular to } \vec{a} \text{ and } \vec{b} =\frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|} = \frac{- \hat{ i } + 2 \hat{ j } + 2 \hat{ k } }{3}$

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APPEARS IN

RD Sharma Class 12 Maths
Chapter 25 Vector or Cross Product
Exercise 25.1 | Q 3.1 | Page 29