1. Understanding Perpendicular Lines
Perpendicular lines are lines that meet or intersect at a right angle (90°). The concept of slope helps us identify whether two lines are perpendicular on the Cartesian plane.
If two lines are perpendicular, one line rises at a certain rate while the other falls at the exact opposite rate, creating a right-angle intersection.
1.1. Definition
Two non-vertical lines are perpendicular if the product of their slopes is -1.
2. Condition for Perpendicular Lines
If two lines have slopes \(m_1\) and \(m_2\), then they are perpendicular if:
2.1. Slope Condition
\( m_1 \cdot m_2 = -1 \)
This shows that the slopes are negative reciprocals of each other.
2.2. Meaning of Negative Reciprocals
Two numbers are negative reciprocals when one is the negative inverse of the other.
- If one slope is \(2\), the perpendicular slope is \(-\dfrac{1}{2}\).
- If one slope is \(-3\), the perpendicular slope is \(\dfrac{1}{3}\).
2.2.1. Geometric Interpretation
A steeper line will have a less steep perpendicular partner, but both meet at 90° when extended.
3. Special Case: Vertical and Horizontal Lines
Vertical and horizontal lines are always perpendicular to each other.
3.1. Vertical Line
A vertical line has undefined slope. Its equation is of the form:
\( x = a \)
3.2. Horizontal Line
A horizontal line has slope 0. Its equation is of the form:
\( y = b \)
3.3. Perpendicular Relationship
Because a vertical line changes x without changing y, and a horizontal line changes y without changing x, the two are perpendicular.
4. Perpendicular Lines in Different Forms of Equations
The slope condition makes it easy to check perpendicularity after converting an equation to a slope-intercept form.
4.1. Slope-Intercept Form Check
Lines:
\( y = m_1x + c_1 \)
\( y = m_2x + c_2 \)
are perpendicular if:
\( m_1 m_2 = -1 \)
4.2. General Form Check
For the form \(Ax + By + C = 0\), the slope is:
\( m = -\dfrac{A}{B} \)
Thus, two lines are perpendicular if:
\( \left( -\dfrac{A_1}{B_1} \right) \left( -\dfrac{A_2}{B_2} \right) = -1 \)
5. Examples
These examples help understand how to check whether two lines are perpendicular.
5.1. Example 1: Using Slopes
Lines:
\( y = 2x + 3 \)
\( y = -\dfrac{1}{2}x + 4 \)
Slopes: 2 and -1/2
\( 2 \cdot -\dfrac{1}{2} = -1 \)
→ Lines are perpendicular.
5.2. Example 2: Using General Form
Lines:
\( 3x + y - 7 = 0 \)
\( x - 3y + 4 = 0 \)
Slopes:
\( m_1 = -3 \)
\( m_2 = \dfrac{1}{3} \)
Product = -1 → Lines are perpendicular.
5.3. Example 3: Vertical and Horizontal
Vertical line: \(x = 5\)
Horizontal line: \(y = -2\)
These lines intersect at 90° → perpendicular.