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If l12+m12+...

Question

If $l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = 1$ , $l_{2}^{2} + m_{2}^{2} + n_{2}^{2} = 1$ ,
$l_{3}^{2} + m_{3}^{2} + n_{3}^{2} = 1$ and $l_{3} l_{1} + m_{3} m_{1} + n_{2} n_{1} = 0$ , then $∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{2} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣$ =

A

0

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B

\pm 1

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C

\pm 2

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D

3

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Solution

The correct option is B $\pm 1$
Consider, $Δ^{2} = ∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{2} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{2} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} l_{1}^{2} + {m_{1}}^{2} + {n_{1}}^{2} & l_{2} l_{1} + m_{2} m_{1} + n_{2} n_{1} & l_{3} l_{1} + m_{3} m_{1} + n_{2} n_{1} l_{2} l_{1} + m_{2} m_{1} + n_{2} n_{1} & l_{2}^{2} + {m_{2}}^{2} + {n_{2}}^{2} & l_{3} l_{2} + m_{3} m_{2} + n_{2} n_{3} l_{3} l_{1} + m_{3} m_{1} + n_{2} n_{1} & l_{3} l_{2} + m_{3} m_{2} + n_{2} n_{3} & l_{3}^{2} + {m_{3}}^{2} + {n_{3}}^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ∣ ∣ ∣ ∣$
$\Rightarrow Δ^{2} = 1$
$\Rightarrow Δ = \pm 1$

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Similar questions

l_{1}, m_{1}, n_{1}; l_{2}, m_{2}, n_{2}; l_{3}, m_{3}, n_{3}

are real quantities satisfying the six relations

l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = l_{2}^{2} + m_{2}^{2} + n_{2}^{2} = l_{3}^{2} + m_{3}^{2} + n_{3}^{2} = 1

l_{2} l_{3} + m_{2} m_{3} + n_{2} n_{3} = l_{3} l_{1} + m_{3} m_{1} + n_{3} n_{1} = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0

∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣ = \pm k

. Find the value of

k

Q. Show the matrix

⎡ ⎢ ⎣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ⎤ ⎥ ⎦

is orthogonal if

l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = \sum l_{1}^{2} = 1 = \sum l_{2}^{2} = \sum l_{3}^{2}

l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = \sum l_{1} l_{2} = 0 = \sum l_{2} l_{3} = \sum l_{3} l_{1}

l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = 1, e t c a n d l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0

Δ = ∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣

Q. If the diagonals of the quadrilateral formed by the lines l₁x + m₁y + n₁ = 0, l₂x + m₂y + n₂ = 0, l₁x + m₁y + n₁' = 0 and l₂x + m₂y + n₂' = 0 are perpendicular, then write the value of l₁² − l₂² + m₁² − m₂².

If the diagonals of the quadrilateral formed by the lines $l - 1 x + m_{1} y + n_{1} = 0, l_{2} x + m_{2} y + n_{2} = 0, l_{1} x + m_{1} y + n_{1}^{'} = 0$ and $(l_{2} x + m_{2} y + n_{2}^{'} = 0$ are perpendicular,then write the value of $l_{1}^{2} - l_{2}^{2} + m_{2} - m_{2}^{2}$

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