Condition for Straight Lines in Plane to be Perpendicular
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Theorem
General Equation
Let $L_1$ and $L_2$ be straight lines embedded in a cartesian plane, given in general form:
\(\ds L_1: \, \) | \(\ds l_1 x + m_1 y + n_1\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds L_2: \, \) | \(\ds l_2 x + m_2 y + n_2\) | \(=\) | \(\ds 0\) |
Then $L_1$ is perpendicular to $L_2$ if and only if:
- $l_1 l_2 + m_1 m_2 = 0$
Slope Form
Let $L_1$ and $L_2$ be straight lines in the Cartesian plane.
Let the slope of $L_1$ and $L_2$ be $\mu_1$ and $\mu_2$ respectively.
Then $L_1$ is perpendicular to $L_2$ if and only if:
- $\mu_1 = -\dfrac 1 {\mu_2}$
Examples
Arbitrary Example $1$
Let $\LL_1$ be the straight line whose equation in general form is given as:
- $3 x - 4 y = 7$
Let $\LL_2$ be the straight line perpendicular to $\LL_1$ which passes through the point $\tuple {1, 2}$.
The equation for $\LL_2$ is:
- $4 x + 3 y = 10$