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Question

If $¯ ¯ ¯ a$ and $¯ ¯ b$ are two non-collinear vectors, then the points $l_{1} ¯ ¯ ¯ a + m_{1} ¯ ¯ b$ , $l_{2} ¯ ¯ ¯ a + m_{2} ¯ ¯ b$ and $l_{3} ¯ ¯ ¯ a + m_{3} ¯ ¯ b$ are collinear if

A

\sum l_{1} (m_{2} - m_{3}) = 0

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B

\sum l_{1} (m_{1} - m_{3}) = 0

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C

\sum l_{1} (m_{1} + m_{3}) = 0

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D

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

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Solution

The correct option is A $\sum l_{1} (m_{2} - m_{3}) = 0$
$¯ a$ and $¯ b$ are non-collinear
$△ = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 l_{1} & l_{2} & l_{3} m_{1} & m_{2} & m_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$
$c_{1} \to c_{1} - c_{2} c_{3} \to c_{1} - c_{3} △ = ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 0 l_{1} - l_{2} & l_{2} & l_{1} - l_{3} m_{1} - m_{2} & m_{2} & m_{1} - m_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$
$△ = - (l_{1} - l_{2}) (m_{1} - m_{3}) + (l_{1} - l_{3}) (m_{1} - m_{2}) = 0$
$\Rightarrow \sum l_{1} (m_{2} - m_{3}) = 0$

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