If $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $(m_{1} n_{2} - m_{2} n_{1}), (n_{1} l_{2} - n_{2} l_{1}),$

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Solution

Solution:
As we know that, $\to a \times \to b$ is perpendicular to both $\to a$ and $\to b$

So, the required line is cross product of lines having direction ratios $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$

Required line $= ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^ı & ^ȷ & ^k l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$

$\Rightarrow^ı (m_{1} n_{2} - m_{2} n_{1}) -^ȷ (l_{1} n_{2} - l_{2} n_{1})$
$+^k (l_{1} m_{2} - l_{2} m_{1})$

$\Rightarrow (m_{1} n_{2} - m_{2} n_{1})^ı + (l_{2} n_{1} - l_{1} n_{2})^ȷ$ $+ (l_{1} m_{2} - l_{2} m_{1})^k$

Hence, the direction cosines of the line perpendicular to both of the given lines are : $(m_{1} n_{2} - m_{2} n_{1}), (n_{1} l_{2} - n_{2} l_{1}), (l_{1} m_{2} - l_{2} m_{1}) .$

Hence proved.

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