# For any two vectors → a and → b , find ( → a × → b ) . → b - Mathematics

Short Note

For any two vectors $\vec{a} \text{ and } \vec{b} , \text{ find } \left( \vec{a} \times \vec{b} \right) . \vec{b} .$

#### Solution

$\text{ Let } :$
$\vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k }$
$\vec{b} = b_1 \hat{ i } + b_2 \hat{ j } + b_3 \hat{ k }$
$\vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix}$
$= \hat{ i } \left( a_2 b_3 - a_3 b_2 \right) - \hat{ j } \left( a_1 b_3 - a_3 b_1 \right) + \hat{ k } \left( a_1 b_2 - a_2 b_1 \right)$
$\left( \vec{a} \times \vec{b} \right) . \vec{b}$
$= \left[ \hat{ i } \left( a_2 b_3 - a_3 b_2 \right) - \hat{ j } \left( a_1 b_3 - a_3 b_1 \right) + \hat{ k } \left( a_1 b_2 - a_2 b_1 \right) \right] . \left( b_1 \hat{ i } + b_2 \hat{ j } + b_3 \hat{ k } \right)$
$= b_1 \left( a_2 b_3 - a_3 b_2 \right) - b_2 \left( a_1 b_3 - a_3 b_1 \right) + b_3 \left( a_1 b_2 - a_2 b_1 \right)$
$= a_2 b_1 b_3 - a_3 b_1 b_2 - a_1 b_2 b_3 + a_3 b_1 b_2 + a_1 b_2 b_3 - a_2 b_1 b_3$
$= 0$

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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 25 Vector or Cross Product
very short answers | Q 13 | Page 33