Introduction
Understanding the angle between two lines is a fundamental concept in geometry and trigonometry. This article explores how to find the angle between two given lines using their slopes and trigonometric functions. Specifically, we will cover an example involving the lines x-2y 1 and x-3y 5.
Background and Theory
The angle between two lines can be determined using the slopes of these lines. The formula to calculate the angle (theta) between two lines with slopes (m_1) and (m_2) is:
[tantheta left| frac{m_1 - m_2}{1 m_1 m_2} right|]
This formula simplifies the process by utilizing the slopes of the lines and a trigonometric function.
Example: Finding the Angle Between Two Lines
Step 1: Convert Equations to Slope-Intercept Form
The first step is to convert the given line equations to their slope-intercept form y mx c.
Line 1: x-2y 1
- Rearrange to slope-intercept form:
-2y -x 1
y frac{1}{2}x - frac{1}{2}
- Slope of Line 1 m_1 frac{1}{2}
Line 2: x-3y 5
- Rearrange to slope-intercept form:
-3y -x 5
y frac{1}{3}x - frac{5}{3}
- Slope of Line 2 m_2 frac{1}{3}
Step 2: Apply the Angle Formula
Now, substitute the slopes m_1 frac{1}{2} and m_2 frac{1}{3} into the formula:
[tantheta left| frac{frac{1}{2} - frac{1}{3}}{1 left(frac{1}{2}right)left(frac{1}{3}right)} right|]
Calculate the numerator:
[frac{1}{2} - frac{1}{3} frac{3}{6} - frac{2}{6} frac{1}{6}]
Calculate the denominator:
[1 left(frac{1}{2}right)left(frac{1}{3}right) 1 frac{1}{6} frac{7}{6}]
Substitute these values back into the formula:
[tantheta left| frac{frac{1}{6}}{frac{7}{6}} right| left| frac{1}{7} right| 1]
Step 3: Determine the Angle
Since tantheta 1, we can find the angle as:
[theta arctan(1) 45°]
Alternative Method: Using Normal Vectors
An alternative method involves using the normal vectors to the lines. For line 1, the normal vector is (1, -(2)) and for line 2, the normal vector is (1, -(3)).
The tangent of the angle between the lines can be calculated using the dot product and magnitudes of the normal vectors:
[tantheta frac{|a_1a_2 b_1b_2|}{|mathbf{N}_1| |mathbf{N}_2|}]
Here, a_1 1, b_1 -2, a_2 1, b_2 -3, and the magnitudes are:
[|mathbf{N}_1| sqrt{1^2 (-2)^2} sqrt{5}]
[|mathbf{N}_2| sqrt{1^2 (-3)^2} sqrt{10}]
Calculate the numerator:
[|1*1 (-2)*(-3)| |1 6| 7]
Substitute these values back into the formula:
[tantheta frac{7}{sqrt{5} cdot sqrt{10}} 1]
Thus, theta 45°
Conclusion
In summary, the angle between the lines x-2y 1 and x-3y 5 is 45 degrees. This demonstrates the application of the slope formula and the alternative method using normal vectors. Both methods yield the same result, providing a robust approach to solving similar problems in geometry and trigonometry.