To find the gradient of a line perpendicular to another line, you invert the original gradient and multiply it by -1.
Understanding the relationship between perpendicular lines is key. As the reference states, "Perpendicular lines have gradients which multiply together to give -1." This fundamental property, often written as m₁ * m₂ = -1 (where m₁ and m₂ are the gradients of the two perpendicular lines), provides the basis for the method.
The Method: Invert and Multiply by -1
Based on the principle m₁ * m₂ = -1 and the explicit instruction from the reference, "To find the gradient of a line which is perpendicular to one of gradient a/b, we invert the fraction and multiply by -1," here are the steps:
- Start with the original gradient: Let's call the original gradient
m. - Express the gradient as a fraction: If the gradient is a whole number (like 5), write it as a fraction over 1 (e.g., 5/1). If it's already a fraction (like 2/3 or -4/5), you're ready.
- Invert the fraction: Flip the numerator and the denominator. For example, if the original gradient is
a/b, the inverted fraction isb/a. - Multiply by -1: Change the sign of the inverted fraction. If it was positive, it becomes negative; if it was negative, it becomes positive. This gives you the perpendicular gradient,
mₚ.
Mathematically, if the original gradient is m, the perpendicular gradient mₚ is found using the formula:
mₚ = -1 / m
This formula directly implements the "invert and multiply by -1" rule.
Examples
Let's look at a few examples, including the one provided in the reference:
-
Example 1 (from reference):
- Original gradient: 5
- Write as fraction: 5/1
- Invert: 1/5
- Multiply by -1: -1/5
- Perpendicular gradient: -1/5
- Check: 5 * (-1/5) = -1. This confirms the relationship.
-
Example 2:
- Original gradient: 2/3
- Write as fraction: (already a fraction) 2/3
- Invert: 3/2
- Multiply by -1: -3/2
- Perpendicular gradient: -3/2
- Check: (2/3) * (-3/2) = -1.
-
Example 3:
- Original gradient: -4/5
- Write as fraction: (already a fraction) -4/5
- Invert: -5/4 (keep the sign for now, or apply it in the next step)
- Multiply by -1: - (-5/4) = 5/4
- Perpendicular gradient: 5/4
- Check: (-4/5) * (5/4) = -1.
Summary Table
| Original Gradient (m) | Inverted Fraction | Multiply by -1 (Perpendicular Gradient mₚ) | Check (m * mₚ) |
|---|---|---|---|
| 5 (or 5/1) | 1/5 | -1/5 | 5 * (-1/5) = -1 |
| 2/3 | 3/2 | -3/2 | (2/3) * (-3/2) = -1 |
| -4/5 | -5/4 | 5/4 | (-4/5) * (5/4) = -1 |
By following the simple rule of inverting the fraction and changing its sign, you can quickly determine the gradient of a line perpendicular to a given line, leveraging the property that their gradients multiply to -1.
