Question

Assertion :Vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar. Reason: $(x\overrightarrow{a}+y\overrightarrow{b})=\overrightarrow{0}\Rightarrow$ x = 0 and y = 0.

• 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 D Assertion is incorrect but Reason is correct

From assertion $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=0$

$\left[\overline{a}\overline{b}\overline{d}\right]\overline{c}-\left[\overline{a}\overline{b}\overline{c}\right]\overline{d}=0$

Assertion $\overline{a},\overline{b},\overline{c}$ are coplanar

Hence $\left[\overline{a}\overline{b}\overline{c}\right]=0$

This does not prove $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=0$ or vise versa

Reason

$x\overrightarrow{a}+y\overrightarrow{b}=0$

It means $x=0,y=0$

$\overrightarrow{a},\overrightarrow{b}$ are not coplanar

$\Rightarrow \overline{a}\ne \lambda \overline{b}$