If a¯ .b¯=a¯ . c¯ , a¯ ×b¯=a¯ × c¯ and a¯ ≠0 then

a¯ . b¯−c¯=0 and a¯ × b¯−c¯=0⇒ either a¯=0 (or) b¯ −c¯=0¯ but a¯ ≠0¯ ⇒ b¯=c¯

If a¯ .b¯=a¯ . c¯ , a¯ ×b¯=a¯ × c¯ and a¯ ≠0 then

a¯ . b¯−c¯=0 and a¯ × b¯−c¯=0⇒ either a¯=0 (or) b¯ −c¯=0¯ but a¯ ≠0¯ ⇒ b¯=c¯

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If $\bar{a} . \bar{b} = \bar{a} . \bar{c}$ , $\bar{a} \times \bar{b} = \bar{a} \times \bar{c}$ and $\bar{a} \neq 0$ then

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$\bar{b} \neq \bar{c}$

$\bar{a} = \bar{b}$

$\bar{b} = \bar{c}$

$\bar{b} = 0$

answer is C.

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Detailed Solution

$\bar{a} . (\bar{b} - \bar{c}) = 0$ and $\bar{a} \times (\bar{b} - \bar{c}) = 0$
$\Rightarrow either \bar{a} = 0$ (or)
$\bar{b} - \bar{c} = \bar{0}$
but $\bar{a} \neq \bar{0} \Rightarrow \bar{b} = \bar{c}$

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If a¯ .b¯=a¯ . c¯ , a¯ ×b¯=a¯ × c¯ and a¯ ≠0 then

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If $\bar{a} . \bar{b} = \bar{a} . \bar{c}$ , $\bar{a} \times \bar{b} = \bar{a} \times \bar{c}$ and $\bar{a} \neq 0$ then