Question

Assertion :Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ be the position vectors of four points $A,B,C$ and $D$ and $3\overrightarrow{a}-2\overrightarrow{b}+5\overrightarrow{c}-6\overrightarrow{d}=\overrightarrow{0}$, then points $A,B,C$ and $D$ are coplanar. Reason: Three non-zero, linearly dependent coinitial vectors $(\overrightarrow{PQ},\overrightarrow{PR}$ and $\overrightarrow{PS})$ are coplanar, then $\overrightarrow{PQ}=\lambda \overrightarrow{PR}+\mu \overrightarrow{PS}$, where $\lambda$ and $\mu$ are scalars.

• 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. Assertion is incorrect but Reason is correct

Answer 1

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

$3\overrightarrow{a}-2\overrightarrow{b}+5\overrightarrow{c}-6\overrightarrow{d}=(2\overrightarrow{a}-2\overrightarrow{b})+(-5\overrightarrow{a}+5\overrightarrow{c})+(6\overrightarrow{a}-6\overrightarrow{d})$
$=-2\overrightarrow{AB}+5\overrightarrow{AC}-6\overrightarrow{AD}=\overrightarrow{0}$
Therefore, $\overrightarrow{AB},\overrightarrow{AC}$ and $\overrightarrow{AD}$ are linearly dependent.
Hence by statement 2, statement 1 is true.