Understanding the Angle Between Two Lines in Coordinate Geometry

In coordinate geometry, the angle between two lines is an important concept that often arises in various applications, from basic geometry to more advanced areas of mathematics. This article will explore how to calculate the angle between the lines given by the equations 4x - y -2 and 6x - y 8. We will examine this using both the vector normal method and the slope method, emphasizing the arctangent function, which plays a key role in determining the angle.

Vector Normal Method

The vector normal method involves the use of vectors perpendicular to the lines. Let's start by identifying the normal vectors for the given lines. A line in the form (ax by c 0) has a normal vector of (vec{n} ai bi).

For the equation (4x - y -2), we can rewrite it as (4x - y 2 0). Thus, the normal vector is (vec{n_1} 4i - j).

For the equation (6x - y 8), we can rewrite it as (6x - y - 8 0). Hence, the normal vector is (vec{n_2} 6i - j).

The angle between the normals can be found using the dot product of the normal vectors, which is defined as:

[ cos(theta) frac{vec{n_1} cdot vec{n_2}}{|vec{n_1}| |vec{n_2}|} ]

First, calculate the dot product of (vec{n_1}) and (vec{n_2}):

[ vec{n_1} cdot vec{n_2} (4i - j) cdot (6i - j) 4 cdot 6 (-1) cdot (-1) 24 1 25 ]

Next, calculate the magnitudes of (vec{n_1}) and (vec{n_2}):

[ |vec{n_1}| sqrt{4^2 (-1)^2} sqrt{16 1} sqrt{17} ] [ |vec{n_2}| sqrt{6^2 (-1)^2} sqrt{36 1} sqrt{37} ]

Now, substitute these values into the dot product formula:

[ cos(theta) frac{25}{sqrt{17} cdot sqrt{37}} frac{25}{sqrt{629}} ]

The angle (theta) can be found using:

[ theta arccos left( frac{25}{sqrt{629}} right) ]

However, the angle between the lines is the supplementary angle to (theta), i.e., (180^circ - theta), leading to:

[ theta 180^circ - arccos left( frac{25}{sqrt{629}} right) ]

Slope Method

Another approach is to use the slopes of the lines. The slope of the line (4x - y -2) is (m_1 -4), and the slope of the line (6x - y 8) is (m_2 6).

The angle (alpha) between two lines with slopes (m_1) and (m_2) can be found using the formula:

[ tan(alpha) left| frac{m_2 - m_1}{1 m_1 m_2} right| ]

Substitute the values of (m_1) and (m_2):

[ tan(alpha) left| frac{6 - (-4)}{1 (-4) cdot 6} right| left| frac{10}{1 - 24} right| left| frac{10}{-23} right| frac{10}{23} ]

Therefore, the angle (alpha) is given by:

[ alpha arctan left( frac{10}{23} right) ]

Conclusion

Both methods provide us with the necessary information to determine the angle between the given lines. Method one, using the vector normal, involved finding the angle between the normals and determining the supplementary angle to get the angle between the lines. Method two, using the slopes, directly computed the angle using the arctangent function. The preferred method often depends on the context and the specific problem at hand.

The calculation using the slopes method yields a more straightforward and commonly used result, often aligning with the arctan function in most applications. This clear and concise approach can be easily understood and applied in various scenarios, making it a practical tool for solving related problems in coordinate geometry.

References

[1] Stewart, J. (2015). Essential Calculus: Early Transcendentals. Cengage Learning.

[2] Kreyszig, E. (2011). Advanced Engineering Mathematics. John Wiley Sons.

[3] Spiegel, M. R. (2008). Schaum's Outline of Theory and Problems of College Mathematics. McGraw-Hill.