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

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?

APPEARS IN

RD Sharma Class 12 Maths
Chapter 25 Vector or Cross Product
MCQ | Q 2 | Page 35