04. Lines

Dated: 30-10-2024

If we have 2 points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) then a line is an object which connects both of these points.
This object has no width or curvature and can also be imagined as a set¹ of points.

Slopes

If \(\overline{AB}\) is a line then, its slope \(m\) is defined to be the ratio from rise to run.

\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac {\Delta y}{\Delta x}\]

Angle of Inclination

From the slope, we can fine the angle of inclination by using \(\tan^{-1}{(m)}\).

\[\tan(\theta) = \frac{\text{perp}}{\text{base}}\]

\[\tan(\theta) = \frac {\Delta y}{\Delta x}\]

\[\tan (\theta) = m\]

\[\theta = \tan^{-1} (m)\]

Parallel and Perpendicular Lines

Parallel

For parallel lines if \(m_1\) and \(m_2\) are slopes of each lines then

\[m_1 = m_2\]

Perpendicular

For perpendicular lines.

\[m_1 = \frac{-1}{m_2}\]

To prove

\[\tan(\theta_1) \cdot \tan(\theta_2) = -1\]

\[\because \theta_2 = \theta_2 + 90^{\circ}\]

\[\tan(\theta_1) \cdot \tan(\theta_1 + 90^\circ) = -1\]

\[\tan(\theta_1) \cdot - \cot(\theta_1 ) = -1\]

The - sign comes from the fact that rise is in \(y^+\) and run is in \(x^-\) .

\[\because \cot(\theta) = \frac{1}{\tan{\theta}}\]

\[\tan(\theta_1) \cdot \frac{-1}{\tan(\theta_1 )} = -1\]

\[-1 = -1\]

Equations of Lines

Lines Parallel to Coordinate Axes

`x-axis`

More generally,

\[x = a\]

Where \(a \in \mathbb{Z}\) .

`y-axis`

If we take \(y\) in terms of function² then notice how it does not depend on \(x\), hence it is constant.

More generally,

\[y = a\]

Where \(a \in \mathbb{Z}\).

Slope - point Form

if we are given slope of a line along with a single point \(P(x_1, y_1)\) which lies on that line then we can construct the equation for it.

\[m = \frac{y - y_1}{x - x_1}\]

\[y - y_1 = m \cdot (x - x_1)\]

Here, the \(P(x, y)\) is any arbitrary point.

Slope - Intercept Form

Relative to `x-axis`

If we have an x intercept called \(a\) then the previous equation takes form of

\[y - 0 = m \cdot (x - a)\]

Relative to `y-axis`

If we have an y intercept called \(b\) then the previous equation takes form of

\[y - b = m \cdot (x - 0)\]

General Equation

\[ax + by + c = 0\]

Where \(a\), \(b\) and \(c\) are constants.
This equation is called first degree equation.