question_answer

If \[\text{l}_{r}^{2}+m_{r}^{2}+n_{r}^{2}=\text{1};\] \[r=1,2,3\] and \[{{\text{l}}_{r}}{{\text{l}}_{s}}+{{m}_{r}}{{m}_{s}}+{{n}_{r}}{{n}_{s}}=0;\]\[r\ne s,\]\[r=1,2,3;\] \[s=1,2,3,\]then the value of \[\left| \begin{matrix}    {{\text{l}}_{1}} & {{m}_{1}} & {{n}_{1}}  \\    {{\text{l}}_{2}} & {{m}_{2}} & {{n}_{2}}  \\    {{\text{l}}_{3}} & {{m}_{3}} & {{n}_{3}}  \\ \end{matrix} \right|\] is

A) \[0\]

B) \[\pm 1\]

C) \[2\]

D) None of these

Correct Answer: B

Solution :

[b] \[{{D}^{2}}=\left| \begin{matrix}    {{l}_{1}} & {{m}_{1}} & {{n}_{1}}  \\    {{l}_{2}} & {{m}_{2}} & {{n}_{2}}  \\    {{l}_{3}} & {{m}_{3}} & {{n}_{3}}  \\ \end{matrix} \right|\left| \begin{matrix}    {{l}_{1}} & {{m}_{1}} & {{n}_{1}}  \\    {{l}_{2}} & {{m}_{2}} & {{n}_{2}}  \\    {{l}_{3}} & {{m}_{3}} & {{n}_{3}}  \\ \end{matrix} \right|\] \[=\left| \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 0  \\    0 & 0 & 1  \\ \end{matrix} \right|=1\Rightarrow D=\pm 1\]