Question

Assertion :If three points $P,Q$ and $R$ have position vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ respectively, and $2\overrightarrow{a}+3\overrightarrow{b}-5\overrightarrow{c}=0$, then the points $P,Q$ and $R$ must be collinear. Reason: If for three points $A,B$ and $C$; $\overrightarrow{AB}=\lambda \overrightarrow{AC}$, then points $A,B$ and $C$ must be collinear.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$2\overrightarrow{a}+3\overrightarrow{b}-5\overrightarrow{c}=0$

$\Rightarrow 3(\overrightarrow{b}-\overrightarrow{a})=5(\overrightarrow{c}-\overrightarrow{a})\Rightarrow \overrightarrow{AB}=\frac{5}{3}\overrightarrow{AC}$

$\Rightarrow \overrightarrow{AB}$ and $\overrightarrow{AC}$ must be parallel since there is a common point $A$. The point $A,B$ and $C$ must be collinear.