equation of parallel and perpendicular lines in general form

Based on the sources and our conversation, the General Form of the equation of a straight line is a powerful representation because it can represent any straight line. This includes vertical lines, which some other forms (like the standard slope-intercept form y = mx + c) cannot represent because their slope is undefined.

The general equation of a line is given by: Ax + By + C = 0

For this equation to represent a line, the coefficients A and B cannot be simultaneously equal to 0.

We can derive conditions for two lines to be parallel or perpendicular using their equations in this general form.

Let’s consider two straight lines represented in the general form: Line 1: A₁x + B₁y + C₁ = 0 Line 2: A₂x + B₂y + C₂ = 0

The sources provide conditions for parallel and perpendicular lines based on the slopes of the lines. For two non-vertical lines with slopes m₁ and m₂:

• They are parallel if and only if their slopes are equal (m₁ = m₂).
• They are perpendicular if and only if the product of their slopes is -1 (m₁ * m₂ = -1).

From the general form Ax + By + C = 0, the slope of a non-vertical line (where B ≠ 0) is given by m = -A/B.

The sources also provide conditions directly using the coefficients of the general form for the case where the y-coefficients (B₁ and B₂) are not zero: Given two straight lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, where b₁, b₂ ̸= 0:

• The lines are parallel to each other, if: a₁ × b₂ = a₂ × b₁ This condition is equivalent to the slopes being equal for non-vertical lines: -a₁/b₁ = -a₂/b₂ implies a₁b₂ = a₂b₁.

• The lines are perpendicular to each other, if: a₁ × a₂ = −b₁ × b₂ This condition is equivalent to the product of the slopes being -1 for non-vertical lines: (-a₁/b₁) * (-a₂/b₂) = -1 implies a₁a₂ / (b₁b₂) = -1, so a₁a₂ = -b₁b₂.

These conditions from source are stated for lines where the ‘b’ coefficients (representing B in Ax+By+C=0) are non-zero, thus applying to non-vertical lines. However, the general form Ax + By + C = 0 itself is capable of representing all lines, including horizontal lines (when A=0) and vertical lines (when B=0).