Straight Lines

Straight Lines:

A straight line is a line that extends infinitely in both directions and has no curves or bends.
The slope-intercept form of a straight line is y = mx + b, where m is the slope of the line and b is the y-intercept.
The point-slope form of a straight line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
The standard form of a straight line is Ax + By = C, where A, B, and C are constants and A and B are not both zero.
Two lines are parallel if and only if they have the same slope and never intersect.

The distance between a point and a line is the length of the perpendicular line segment from the point to the line.
To find the distance between a point and a line, you can use the formula d = |Ax1 + By1 + C| / sqrt(A^2 + B^2), where (x1, y1) is the point and Ax + By + C = 0 is the equation of the line.
The sign of the expression Ax1 + By1 + C indicates which side of the line the point is on.
If the line passes through the origin (0,0), the distance between the point and the line is simply |Ax1 + By1| / sqrt(A^2 + B^2).
The shortest distance between two parallel lines is the perpendicular distance between them, which can be found using the same formula as above.

Cartesian form: $$\frac{|ax + by + c|}{\sqrt{a^2 + b^2}}$$
Vector form: $$\frac{|\mathbf{n} \cdot \mathbf{p} + d|}{\lVert \mathbf{n} \rVert}$$

The angle between two lines is the angle formed by their intersection.
The slope of a line is related to the angle it makes with the positive x-axis by the formula theta = arctan(m), where m is the slope.
The angle between two lines with slopes m1 and m2 is given by the formula theta = |arctan((m1-m2) / (1+m1m2))|.
The angle between two perpendicular lines is 90 degrees or pi/2 radians.
The angle between two parallel lines is either 0 degrees or pi radians.

Formula: $$\theta = \cos^{-1}\left(\frac{m_1m_2 - 1}{\sqrt{1 + m_1^2}\sqrt{1 + m_2^2}}\right)$$

Two non-parallel lines will intersect at a single point, unless they are coincident (i.e., the same line).
The point of intersection can be found by solving the system of equations Ax + By = C and Dx + Ey = F simultaneously.
If the lines are given in slope-intercept form, you can solve for the intersection point by setting their y-values equal to each other and solving for x, then plugging that value back into either equation to find the corresponding y-value.
If the lines are given in point-slope form, you can find the intersection point by plugging the coordinates of one of the points into the equation for the other line and solving for the parameter.
If the lines are given in standard form, you can solve for the intersection point by finding the determinant of a 2x2 matrix formed from the coefficients of x and y in the two equations.

Formula: $$\begin{aligned} x &= \frac{b_2c_1 - b_1c_2}{a_1b_2 - a_2b_1} \\ y &= \frac{c_2a_1 - c_1a_2}{a_1b_2 - a_2b_1} \end{aligned}$$

Two parallel lines have the same slope and will never intersect.
The distance between two parallel lines is the length of the shortest line segment that is perpendicular to both lines.
To find the distance between two parallel lines, you can find the distance between one of the lines and any point on the other line.
Alternatively, you can use the distance formula for a point and a line to find the distance between a point on one line and the other line.
The distance between two parallel lines is the absolute value of the difference between their y-intercepts, divided by the square root of the sum of the squares of their slopes.