Derive the perpendicularity between two lines for the relation $m_{1} \times m_{2} = - 1$

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Solution

To understand why $m_{1} m_{2} = - 1$ , does not apply to x-axis andy-axis, we ust know how $m_{1} m_{2} = - 1$ derived.
Let $l_{1}$ and $l_{2}$ be 2 lines with angles of inclination with x-axis $a, θ_{1} a n d θ_{2}$
Let $θ$ be the angle between these lines
$θ = θ_{2} - θ_{1}$
$t a n θ = t a n (θ_{1} - θ_{1}) = \frac{t a n θ_{2} - t a n θ_{1}}{1 + t a n θ_{1} t a n θ_{2}}$
$\begin{matrix} B u t t a n θ_{1} = m_{1} (s l o p e o f l_{1}) a n d t a n θ_{2} = m_{2} (s l o p e o f l_{2}) \end{matrix}} \to t a n θ = \frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$
If lines are $⊥$ to each other $t a n θ = t a n 90^{\circ} = \infty$
$∴$ Denominator is zero $\Rightarrow 1 + m_{1} m_{2} = 0$
or $m_{1} m_{2} = - 1$
$\begin{matrix} F o r x - a x i s m_{1} = 0 y - a x i s m_{2} = \infty \end{matrix}} \to t a n θ = \frac{\infty - 0}{1 + 0 o r \infty} = n o t d e f i n e d$
Hence, $m_{1} m_{2} = 1$ is not applicable

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Derive the perpendicularity between two lines for the relation m1×m2=−1

Derive the perpendicularity between two lines for the relation $m_{1} \times m_{2} = - 1$