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Which of the following conclusion(s) hold(s) true for a non-zero vector $\vec{a} ?$

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$\vec{a} . \vec{b} = \vec{a} . \vec{c} \Rightarrow \vec{b} = \vec{c}$

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

$\vec{a} . \vec{b} = \vec{a} . \vec{c} a n d \vec{a} \times \vec{b} = \vec{a} \times \vec{c} \Rightarrow \vec{b} = \vec{c}$

$| \hat{a} + \hat{b} | = | \hat{a} - \hat{b} | \Rightarrow \hat{a} . \hat{b} \neq 0$

answer is C.

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Detailed Solution

$A . \vec{a} . \vec{b} = \vec{a} . \vec{c} \Rightarrow e i t h e r \vec{b} = \vec{c} o r \vec{a} ⊥ \vec{b} - \vec{c}$

$B . \vec{a} \times \vec{b} = \vec{a} \times \vec{c} \Rightarrow e i t h e r \vec{b} = \vec{c} o r \vec{a} ∥ \vec{b} - \vec{c}$

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