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If → a ⋅ → B = → a ⋅ → C and → a × → B = → a × → C , → a ≠ 0 , Then - Mathematics

MCQ

If \[\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}\] and \[\vec{a} \times \vec{b} = \vec{a} \times \vec{c,} \vec{a} \neq 0,\] then

Options

  • \[\vec{b} = \vec{c}\]

  • \[\vec{b} = \vec{0}\]

  • \[\vec{b} + \vec{c} = \vec{0}\]

  • none of these

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Solution

\[\vec{b} = \vec{c}\]

\[\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} \]

\[ \Rightarrow \vec{a}^{} \cdot \vec{b} - \vec{a} \cdot \vec{c} = 0\]

\[ \Rightarrow \vec{a .} \left( \vec{b} - \vec{c} \right) = 0 \]

\[\text { Let } \theta \text { be the angle between} \ \vec{ a } \text { and }\left( \vec{b} - \vec{c} \right) \]

\[\left| \vec{a} \right|\left| \left( \vec{b} - \vec{c} \right) \right|\cos \theta . . . (1)\]

\[\text { and } \vec{a} \times \vec{b} = \vec{a} \times \vec{c} \]

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

\[ \Rightarrow \vec{a} \times \left( \vec{b} - \vec{c} \right) = 0\]

\[\text { Then } , \left| \vec{a} \right| \left| \left( \vec{b} - \vec{c} \right) \right| \sin \theta = 0 . . . (2)\]

\[\text { Here, it is given that} \ \vec{a} \neq 0\]

\[\text { Therefore, for eq (1) and eq (2) to be 0 }\]

We have , 

\[\left| \left( \vec{b} - \vec{c} \right) \right| \cos \theta = 0 \]

\[\text { For } \left| \left( \vec{b} - \vec{c} \right) \right| \cos \theta = 0 , \text { one of } \left| \left( \vec{b} - \vec{c} \right) \right| \text { or }\cos \theta \text { must be } 0\]

Case 1: 

\[\text { Let } \cos \theta = 0\]

\[ \Rightarrow \theta = 90^\circ \]

\[ \Rightarrow \sin \theta = 1\]

\[\text { & if } \left| \left( \vec{b} - \vec{c} \right) \right| \sin \theta = 0 \text { and } \sin \theta = 1 \]

\[\text { Then } \left| \left( \vec{b} - \vec{c} \right) \right| = 0\]

\[ \Rightarrow \vec{b} = \vec{c} \]

Case 2: 

\[\text { Let } \left| \left( \vec{b} - \vec{c} \right) \right| = 0\]

\[ \Rightarrow \vec{b} = \vec{c} \]

\[\text { Hence }, \vec{b} = \vec{c} \]

  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
MCQ | Q 2 | Page 35
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