For any three non-zero vectors $${{r}_{1}},\,{{r}_{2}}$$ and $${{r}_{3}}$$, $$\left| \,\begin{matrix} {{r}_{1}}\,.\,{{r}_{1}} & {{r}_{1}}\,.\,{{r}_{2}} & {{r}_{1}}\,.\,{{r}_{3}} \ {{r}_{2}}\,.\,{{r}_{1}} & {{r}_{2}}\,.\,{{r}_{2}} & {{r}_{2}}\,.\,{{r}_{3}} \ {{r}_{3}}\,.\,{{r}_{1}} & {{r}_{3}}\,.\,{{r}_{2}} & {{r}_{3}}\,.\,{{r}_{3}} \ \end{matrix} \right|=0$$. Then which of the following is false [AMU 2000]

Question

For any three non-zero vectors $${{r}_{1}},\,{{r}_{2}}$$ and $${{r}_{3}}$$, $$\left| \,\begin{matrix}    {{r}_{1}}\,.\,{{r}_{1}} & {{r}_{1}}\,.\,{{r}_{2}} & {{r}_{1}}\,.\,{{r}_{3}}  \    {{r}_{2}}\,.\,{{r}_{1}} & {{r}_{2}}\,.\,{{r}_{2}} & {{r}_{2}}\,.\,{{r}_{3}}  \    {{r}_{3}}\,.\,{{r}_{1}} & {{r}_{3}}\,.\,{{r}_{2}} & {{r}_{3}}\,.\,{{r}_{3}}  \ \end{matrix} \right|=0$$. Then which of the following is false                                    [AMU 2000]

TuteeHUB · Accepted Answer

All the three vectors are linearly dependent

TuteeHUB · Answer

All the three vectors are parallel to one and the same plane

TuteeHUB · Answer

This system of equation has a non-trivial solution

TuteeHUB · Answer

All the three vectors are perpendicular to each other

1.	For any three non-zero vectors \[{{r}_{1}},\,{{r}_{2}}\] and \[{{r}_{3}}\], \[\left\| \,\begin{matrix} {{r}_{1}}\,.\,{{r}_{1}} & {{r}_{1}}\,.\,{{r}_{2}} & {{r}_{1}}\,.\,{{r}_{3}} \\ {{r}_{2}}\,.\,{{r}_{1}} & {{r}_{2}}\,.\,{{r}_{2}} & {{r}_{2}}\,.\,{{r}_{3}} \\ {{r}_{3}}\,.\,{{r}_{1}} & {{r}_{3}}\,.\,{{r}_{2}} & {{r}_{3}}\,.\,{{r}_{3}} \\ \end{matrix} \right\|=0\]. Then which of the following is false [AMU 2000]
A.	All the three vectors are parallel to one and the same plane
B.	All the three vectors are linearly dependent
C.	This system of equation has a non-trivial solution
D.	All the three vectors are perpendicular to each other
Answer» B. All the three vectors are linearly dependent

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