Question

Assertion :The points with position vectors $\overline{a},\overline{b},\overline{c}$ are collinear if $\overline{a}\times \overline{b}+\overline{b}\times \overline{c}+\overline{c}\times \overline{a}=\overline{0}$ Reason: Three vectors are collinear iff $\overrightarrow{AB}=\lambda \overrightarrow{AC}$ where $\lambda ϵR$

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

Let $A,B,C$ are three points with position vectors $\overline{a},\overline{b},\overline{c}$ respectively
$\therefore \overrightarrow{AB}=PositionvectorofB-PositionvectorofA=\overline{b}-\overline{a}$
$\overrightarrow{AC}=PositionvevtorofC-PositionvectorofA=\overrightarrow{c}-\overrightarrow{a}$
Now $\overrightarrow{AB}\parallel \overrightarrow{AC}$
$\Rightarrow \overrightarrow{AB}\times \overrightarrow{AC}=0$ $\Rightarrow (\overline{b}-\overline{a})\times (\overrightarrow{c}-\overrightarrow{a})=0$
$\Rightarrow \overline{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overline{c}\times \overrightarrow{a}=\overrightarrow{0}$
$\Rightarrow$ Assertion (A) & reason (R) both are true & reason (R) is correct explanation of assertion (A)