The Condition for Two Lines to be Parallel or Perpendicular

Understanding the Concept

Two straight lines are parallel when the lines extend indefinitely without meeting. In a Cartesian coordinate system, lines are parallel if their gradients are the same. So if lines L1 and L2 with gradients m1 and m2 respectively are parallel, then m1 = m2.
Perpendicular lines are lines that intersect at a right angle (90 degrees). In the Cartesian plane, two lines are perpendicular if the product of their gradients equals -1. Hence, for lines L1 and L2 with gradients m1 and m2 respectively are perpendicular, then m1 * m2 = -1.

The conditions for parallelism and perpendicularity can be applied to check whether given lines are parallel or perpendicular. Additionally, in the case when the gradient of one line and a point through which the second line passes are given, the gradient of the second line (whether parallel or perpendicular) can be determined.
For instance, if you are given a line L1 with gradient m1 = 2 and you need to find a line parallel to L1 that passes through (-1,3), you can determine the equation using the principle of parallel lines and the equation of a line in slope-point form: y - y1 = m(x - x1).
Similarly, if you want to find a line perpendicular to L1 that passes through (-1,3), you would use -1/2 as your gradient in the slope-point equation because the product m1 * m2 = -1, giving m2 = -1/m1 = -1/2.

To check if the lines given by the equations 2x - 3y + 4 = 0 and 4x - 6y + 5 = 0 are parallel, you would find the gradients of both lines. In this case, both lines have the gradient -2/3. Hence, these lines are parallel lines because their gradients are equal.
To examine if the lines given by the equations 2x + 4y - 6 = 0 and 4y - 2x + 1 = 0 are perpendicular, you would find the gradients of both lines. In this case, line 1 has a gradient of -1/2 and line 2 has a gradient of 2. These lines are perpendicular because their gradient product equals -1.

The gradients of two lines offer valuable information about the orientation of the lines: if the gradients are the same, the lines are parallel; if the gradient product is -1, the lines are perpendicular.
A solid grasp of these facts and being comfortable with working out gradients and rearranging equations are crucial for mastering topics in coordinate geometry.