Question

Assertion :If $\overrightarrow{a}=\hat{i},\overrightarrow{b}=\hat{j}$ and $\overrightarrow{c}=\hat{i}+\hat{j}$ then $\overrightarrow{a}$ and $\overrightarrow{b}$ are linearly independent but $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are linearly dependent. Reason: If $\overrightarrow{a}$ and $\overrightarrow{b}$ are linearly dependent and $\overrightarrow{c}$ is any vector then $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are linearly dependent.

• 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 C Assertion is correct but Reason is incorrect

Given vector

$\overrightarrow{a}=\hat{i}$

$\overrightarrow{b}=\hat{j}$

$\overrightarrow{c}=\hat{i}+\hat{j}$

to check linearly dependent or not

$\overrightarrow{a}=\lambda \overrightarrow{b}$

$\hat{i}=\lambda \hat{j}$

on comparing both sides

$\lambda =0$

HERE we can say by value of $\lambda$ that $\overrightarrow{a}and\overrightarrow{b}$ are linearly independent

$\overrightarrow{a}={\lambda }_1\overrightarrow{b}+{\lambda }_2\overrightarrow{c}$

$\hat{i}={\lambda }_1\hat{j}+{\lambda }_2\hat{i}+{\lambda }_2\hat{j}$

$\hat{i}={\lambda }_2\hat{i}+({\lambda }_1+{\lambda }_2)\hat{j}$

on comparing

${\lambda }_2=1$

${\lambda }_1+{\lambda }_2=0$

${\lambda }_1=-{\lambda }_2$

i.e. a,b,c are linearly dependent