# The Value of ( → a × → B ) 2 is - Mathematics

MCQ

The value of $\left( \vec{a} \times \vec{b} \right)^2$ is

#### Options

• $\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 - \left( \vec{a} \cdot \vec{b} \right)^2$

• $\left| \vec{a} \right|^2 \left| \vec{b} \right|^2 - \left( \vec{a} \cdot \vec{b} \right)^2$

• $\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 - 2\left( \vec{a} \cdot \vec{b} \right)$

• $\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 - \vec{a} \cdot \vec{b}$

#### Solution

$\left( \vec{a} . \vec{b} \right)^2 + \left| \vec{a} \times \vec{b} \right|^2$
$= \left( \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta \right)^2 + \left( \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta \right)^2$
$= \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \left( \cos^2 \theta + \sin^2 \theta \right)$
$= \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \left( 1 \right)$
$= \left| \vec{a} \right|^2 \left| \vec{b} \right|^2$
$\therefore \left| \vec{a} \times \vec{b} \right|^2 = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 - \left( \vec{a} . \vec{b} \right)^2$
$\text{ Thus, the value of } \left( \vec{a} \times \vec{b} \right)^2 \text{ is } \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 - \left( \vec{a} . \vec{b} \right)^2 .$

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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 25 Vector or Cross Product
MCQ | Q 13 | Page 36