Question

If a_i² + b_i² + c_i² = 1(i = 1,2,3) and a_i a_j + b_i b_j + c_i c_j = 0(i ¹ j, i, j = 1,2,3) , then value of $\left| \begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} \right|^{2}$ is

• A. 0
• B. ½
• C. 1
• D. 2

Answer 1

Answer: (C)

1

Answer 2

$\left| \begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} \right|^{2}$= $\left| \begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} \right|$ $\left| \begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} \right|$

=$\left| \begin{matrix} a_{1}^{2} + b_{1}^{2} + c_{1}^{2} & a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} & a_{1}a_{3} + b_{1}b_{3} + c_{1}c_{3} \\ a_{2}a_{1} + b_{2}b_{1} + c_{2}c_{1} & a_{2}^{2} + b_{2}^{2} + c_{2}^{2} & a_{2}a_{3} + b_{2}b_{3} + c_{2}c_{3} \\ a_{3}a_{1} + b_{3}b_{1} + c_{3}c_{1} & a_{3}a_{2} + b_{3}b_{2} + c_{3}c_{2} & a_{3}^{2} + b_{3}^{2} + c_{3}^{2} \end{matrix} \right|$

= $\left| \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right|$ = 1