Question

Assertion :Given 3 vectors
${\overline{V}}_1=a\hat{i}+b\hat{j}+c\hat{k}$ ; ${\overline{V}}_2=b\hat{i}+c\hat{j}+a\hat{k}$ ; ${\overline{V}}_3=c\hat{i}+a\hat{j}+b\hat{k}$
where a,b,c are distinct positive real numbers.
Statement-1:${\overline{V}}_1,{\overline{V}}_2$ and ${\overline{V}}_3$ are linearly dependent vectors.
because Reason: Statement -2: $\left[{\overline{V}}_1{\overline{V}}_2{\overline{V}}_3\right]\ne 0.$

• A. Statement $-$ 1 is true, Statement $-$ 2 is true and Statement $-$ 2 is correct explanation for Statement $-$ 1.
• B. Statement $-$ 1 is true, Statement $-$ 2 is true and Statement $-$ 2 is NOT the correct explanation for Statement $-$ 1.
• C. Statement $-$ 1 is true, Statement $-$ 2 is false.
• D. Statement $-$ 1 is false, Statement $-$ 2 is true.

Answer 1

The correct option is D Statement $-$ 1 is false, Statement $-$ 2 is true.

Given

$\Rightarrow \overrightarrow{V_1}=a\hat{i}+b\hat{j}+c\hat{k}$

$\Rightarrow \overrightarrow{V_2}=b\hat{i}+c\hat{j}+a\hat{k}$

$\Rightarrow \overrightarrow{V_3}=c\hat{i}+a\hat{j}+b\hat{k}$

$\Rightarrow \left[{\overrightarrow{V}}_1{\overrightarrow{V}}_2{\overrightarrow{V}}_3\right]=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

$\Rightarrow \left[{\overrightarrow{V}}_1{\overrightarrow{V}}_2{\overrightarrow{V}}_3\right]=a(cb-a^2)-b(b^2-ca)+c(ba-c^2)$

$\Rightarrow \left[{\overrightarrow{V}}_1{\overrightarrow{V}}_2{\overrightarrow{V}}_3\right]=acb-a^3-b^3-cab+cba-c^3$

$\Rightarrow \left[{\overrightarrow{V}}_1{\overrightarrow{V}}_2{\overrightarrow{V}}_3\right]=acb-(a^3+b^3+c^3)\ne 0$

So $\Rightarrow \overrightarrow{V_1},\overrightarrow{V_2},\overrightarrow{V_3}$ are not linearly dependent vectors.