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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.

 
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Solution

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

\[ \Rightarrow \vec{a} \times \vec{b} = \vec{0} \]

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

\[\text{ For example } :\]

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

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

\[\text{ Here } ,\]

\[ \vec{a} \neq0\]

\[ \vec{b} \neq0\]

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

\[ = 0 \hat{ i }  + 0 \hat{ j }  + 0 \hat{ k }  \]

\[ = \vec{0}\]

  Is there an error in this question or solution?
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APPEARS IN

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