Angle between two lines
Angle Between Two Lines
Understanding the angle between two lines is a fundamental concept in geometry, particularly when dealing with straight lines in a plane. When two non-parallel lines intersect, they form an angle at the point of intersection. This angle is a measure of the separation between the two lines.
Formula for Angle Between Two Lines
The angle between two lines can be calculated using the slopes of the lines. If the slopes of the two lines are $m_1$ and $m_2$, the tangent of the angle $\theta$ between the two lines is given by the following formula:
[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| ]
It is important to note that this formula gives the acute angle between the two lines. If you need the obtuse angle, you can calculate it as $\pi - \theta$ (in radians) or $180^\circ - \theta$ (in degrees).
Important Points
| Point | Description |
|---|---|
| Slope of a Line | The slope of a line, often denoted as $m$, is a measure of its steepness and is calculated as the ratio of the change in y to the change in x. |
| Acute Angle | The angle less than $90^\circ$ (or $\frac{\pi}{2}$ radians) between two lines. |
| Obtuse Angle | The angle greater than $90^\circ$ (or $\frac{\pi}{2}$ radians) and less than $180^\circ$ (or $\pi$ radians) between two lines. |
| Perpendicular Lines | When two lines are perpendicular, their slopes are negative reciprocals of each other ($m_1 = -\frac{1}{m_2}$), and the angle between them is $90^\circ$ (or $\frac{\pi}{2}$ radians). |
| Parallel Lines | Parallel lines have the same slope ($m_1 = m_2$), and the angle between them is $0^\circ$. |
Examples
Example 1: Acute Angle Between Two Lines
Consider two lines with slopes $m_1 = 2$ and $m_2 = \frac{1}{2}$. To find the acute angle $\theta$ between them, we use the formula:
[ \tan(\theta) = \left| \frac{2 - \frac{1}{2}}{1 + 2 \cdot \frac{1}{2}} \right| = \left| \frac{\frac{3}{2}}{\frac{5}{2}} \right| = \left| \frac{3}{5} \right| ]
Now, we find the angle $\theta$ such that $\tan(\theta) = \frac{3}{5}$. Using a calculator or trigonometric tables, we find that $\theta \approx 30.96^\circ$.
Example 2: Obtuse Angle Between Two Lines
Let's find the obtuse angle between the same lines from Example 1. Since we already know the acute angle is $\theta \approx 30.96^\circ$, the obtuse angle $\phi$ is:
[ \phi = 180^\circ - \theta \approx 180^\circ - 30.96^\circ \approx 149.04^\circ ]
Example 3: Perpendicular Lines
If we have two lines with slopes $m_1 = 3$ and $m_2 = -\frac{1}{3}$, these lines are perpendicular to each other because $m_1 = -\frac{1}{m_2}$. The angle between them is $90^\circ$.
Example 4: Parallel Lines
For two lines with slopes $m_1 = m_2 = 4$, the lines are parallel, and the angle between them is $0^\circ$.
Practice Problems
- Find the acute angle between the lines with slopes $m_1 = -1$ and $m_2 = 3$.
- Determine the obtuse angle between the lines with equations $y = 2x + 5$ and $y = -\frac{1}{2}x - 3$.
- Are the lines with equations $4y - 2x = 6$ and $2y + x = 1$ perpendicular? Why or why not?
By understanding the concept of the angle between two lines and using the formula provided, you can solve various problems related to the orientation and intersection of lines in a plane. This knowledge is particularly useful in fields such as geometry, trigonometry, and calculus.