# If Either → a = → 0 Or → B = → 0 , Then → a × → B = → 0 . is the Converse True? Justify Your Answer with an Example. - Mathematics

Sum

If either  $\vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0} , \text{ then } \vec{a} \times \vec{b} = \vec{0} .$  Is the converse true? Justify your answer with an example.

#### Solution

$\text{ If } \vec{a} = \vec{0} \text{ or } \vec{b} =0, \text{ then } \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta \hat{ n } = \vec{0 .}$

$\Rightarrow \vec{a} \times \vec{b} = \vec{0}$

$\text{ But the converse is not true as whenever } \vec{a} \times \vec{b} = \vec{0} , \text{ we cannot be sure that either } \vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0} .$

$\text{ For example } :$

$\vec{a} = \hat{ i } + 2 \hat{ j } + 3 \hat{ k }$

$\vec{b} = \hat{ i } + 2 \hat{ j } + 3 \hat{ k }$

$\text{ Here } ,$

$\vec{a} \neq0$

$\vec{b} \neq0$

$\text{ But } \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 1 & 2 & 3 \\ 1 & 2 & 3\end{vmatrix}$

$= 0 \hat{ i } + 0 \hat{ j } + 0 \hat{ k }$

$= \vec{0}$

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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 25 Vector or Cross Product
Exercise 25.1 | Q 32 | Page 31