Understanding the Angle Between Two Straight Lines: A Guide

In this article, we will explore the process of finding the angle between two given straight lines. We'll approach this problem using two methods: vector equations and slope-intercept form. By the end, you'll have a robust understanding of this topic and be able to apply the same methods to other problems.

Introduction to the Problem

We are asked to find the angle between the two straight lines defined by the equations:

y - √3x - 5 0

√3y - x - 6 0

Method: Vector Equation

First, we'll convert these equations into vector form to find the angle between them.

Step 1: Convert to Vector Form

The first line can be written as:

y - √3x - 5 0 implies sqrt{3}x y - 5 0

The directional vector for this line is:

vec{a} begin{pmatrix} 1 sqrt{3} end{pmatrix}

For the second line:

sqrt{3}y - x - 6 0 implies -x sqrt{3}y - 6 0

The directional vector for this line is:

vec{b} begin{pmatrix} -1 sqrt{3} end{pmatrix}

Step 2: Calculate the Angle Using the Dot Product

The formula to find the angle between two vectors is:

$cos theta frac{vec{a} cdot vec{b}}{|vec{a}| cdot |vec{b}|}$

Calculating the dot product:

$vec{a} cdot vec{b} (-1) cdot 1 sqrt{3} cdot sqrt{3} -1 3 2sqrt{3}$

Calculating the magnitudes:

$|vec{a}| sqrt{1^2 sqrt{3}^2} sqrt{1 3} 2$ $|vec{b}| sqrt{(-1)^2 sqrt{3}^2} sqrt{1 3} 2$

Substituting these values into the formula:

$cos theta frac{2sqrt{3}}{2 cdot 2} frac{sqrt{3}}{2}$

Thus:

$theta frac{pi}{6}$ or $30°$

Method: Slope-Intercept Form

In this method, we convert the given equations to slope-intercept form and then find the angle between the lines.

Step 1: Convert to Slope-Intercept Form

The first line:

$y - √3x - 5 0 implies y √3x 5$

Slope of the line: $m_1 √3$

The second line:

$√3y - x - 6 0 implies √3y x 6 implies y frac{1}{√3}x 2$

Slope of the line: $m_2 frac{1}{√3}$

Step 2: Calculate the Angle Between the Lines

Using the formula for the angle between two lines with slopes $m_1$ and $m_2$:

$tan theta left| frac{m_1 - m_2}{1 m_1 cdot m_2} right|$

Substituting the values:

$tan theta left| frac{√3 - frac{1}{√3}}{1 √3 cdot frac{1}{√3}} right| left| frac{frac{3 - 1}{√3}}{2} right| frac{1}{√3}$

Taking the inverse tangent (arctan) of both sides:

$theta arctan left(frac{1}{√3}right) 30°$

Since the obtuse angle is given by:

$180° - 30° 150°$

Key Points to Remember

1. **Vector Equations**: Representing lines in vector form allows us to utilize the dot product and magnitudes to find the angle between them.

2. **Slope-Intercept Form**: Converting lines to slope-intercept form makes it straightforward to apply the angle formula for two lines.

3. **Angle Calculation**: Both methods yield the same acute angle of 30° and the obtuse angle of 150°.

Conclusion

By understanding both vector and slope-based methods, we can confidently find the angle between any two lines. This knowledge is crucial for various applications in geometry, physics, and engineering.