Assertion (A) : Three points with position vectors a, b and c are… - Vidyadip

Question

Assertio$n (A) :$ Three points with position vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are collinear if $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\overrightarrow{0}$
Reason $(R):$ If $\overrightarrow{A B} \cdot \overrightarrow{A C}=0$, then $\overrightarrow{A B} \perp \overrightarrow{A C}$.

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Answer

(b) : If $A, B, C$ are collinear, then $\overrightarrow{A B}=k \overrightarrow{A C}$
$\therefore \overrightarrow{A B} \times \overrightarrow{A C}=\overrightarrow{0} \Rightarrow(\vec{b}-\vec{a}) \times(\vec{c}-\vec{a})=\overrightarrow{0}$
$\Rightarrow \vec{b} \times \vec{c}+\vec{a} \times \vec{b}+\vec{c} \times \vec{a}=\overrightarrow{0} \text { i.e., } \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\overrightarrow{0}$
Hence, both assertion and reason are true but reason is not the correct explanation of assertion.

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