Check whether the pair of lines $x + 3 y - 7 = 0$ and $3 x - y - 21 = 0$ is perpendicular or not.

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Solution

Step 1: Finding the slope of the line $x + 3 y - 7 = 0$ :
The equation of the line is $x + 3 y - 7 = 0$
$\Rightarrow 3 y = - x + 7$
$\Rightarrow y = \frac{- 1}{3} x + \frac{7}{3}$
Comparing with $y = m x + c$
$\Rightarrow m_{1} = \frac{- 1}{3}$ (where $m_{1}$ is the slope of the line)
Step2: Finding the slope of the line $3 x - y - 21 = 0$ :
The equation of the line is $3 x - y - 21 = 0$
$\Rightarrow y = 3 x - 21$
Comparing with $y = m x + c$
$\Rightarrow m_{2} = 3$ (where $m_{2}$ is the slope of the line)
Step 3: Determining whether the given lines are perpendicular:
We know that, $m_{1} = \frac{- 1}{3}$ and $m_{2} = 3$
Consider, $m_{1} . m_{2} = \frac{- 1}{3} \times 3$
$\Rightarrow m_{1} . m_{2} = - 1$
This shows the lines are perpendicular.
Hence, the two given lines are perpendicular.

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