To calculate the perpendicular gradient of a line, you first need to know the gradient of the original line. The key principle is that the product of the gradients of two perpendicular lines is always -1.
Here's a step-by-step guide:
- Identify the Gradient of the Original Line: Let's call this gradient m1. You may be given this directly or have to calculate it from two points on the line using the formula: m1 = (y2 - y1) / (x2 - x1).
- Calculate the Perpendicular Gradient: If m2 is the gradient of the line perpendicular to the first line, then m1 × m2 = -1. To find m2, rearrange this formula: m2 = -1 / m1.
- This means the perpendicular gradient is the negative reciprocal of the original gradient.
Understanding the Concept
- Reciprocal: The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 2 is 1/2, and the reciprocal of -3/4 is -4/3.
- Negative Reciprocal: You find the negative reciprocal by first finding the reciprocal and then changing its sign.
Examples
Let's explore a few examples:
| Original Gradient (m1) | Perpendicular Gradient (m2) | Calculation |
|---|---|---|
| 2 | -1/2 | m2 = -1 / 2 |
| -3/4 | 4/3 | m2 = -1 / (-3/4) = 4/3 |
| -1 | 1 | m2 = -1 / -1 = 1 |
| 1/5 | -5 | m2 = -1 / (1/5) = -5 |
Practical Insights
- If you have the equation of a line in the form y = mx + c, the coefficient m is the gradient.
- Understanding perpendicular gradients is crucial in geometry and coordinate geometry, particularly when dealing with right-angled triangles and finding the equation of a line perpendicular to a given line.
Conclusion
The key takeaway is that to find the gradient of a line perpendicular to another, you must calculate the negative reciprocal of the original gradient. This simple rule is based on the relationship m1 × m2 = -1, where m1 is the gradient of the original line and m2 is the gradient of the perpendicular line.
