Angle Between Lines

Angle Between Lines#

Slope of a Line Revision#

Theory#

Before finding angles between lines, let’s review the concept of slope:

  • Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for a line through \((x_1, y_1)\) and \((x_2, y_2)\)

  • For line \(ax + by + c = 0\), slope \(m = -\frac{a}{b}\)

  • Parallel lines have equal slopes

  • Perpendicular lines have slopes whose product is \(-1\)

Application#

For the line through \(A(2, 3)\) and \(B(5, 9)\):

  • Slope \(m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2\)

Angle Between Two Lines#

Theory#

The angle between two lines can be found using their slopes or direction vectors.

Using Slopes#

For two lines with slopes \(m_1\) and \(m_2\), the acute angle \(\theta\) between them is: $\(\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1m_2}\right|\)$

Special Cases#

  1. Parallel Lines: If \(m_1 = m_2\), then \(\theta = 0°\)

  2. Perpendicular Lines: If \(m_1m_2 = -1\), then \(\theta = 90°\)

  3. Vertical Line: Use direction vectors instead

Using Direction Vectors#

For lines with direction vectors \(\vec{v_1} = (a_1, b_1)\) and \(\vec{v_2} = (a_2, b_2)\): $\(\cos \theta = \frac{|\vec{v_1} \cdot \vec{v_2}|}{|\vec{v_1}||\vec{v_2}|} = \frac{|a_1a_2 + b_1b_2|}{\sqrt{a_1^2 + b_1^2}\sqrt{a_2^2 + b_2^2}}\)$

Interactive Visualization: Angle Between Lines Explorer#

Application#

Examples#

Example 1: Angle Between Lines with Given Slopes Find the angle between lines with slopes \(m_1 = 2\) and \(m_2 = -\frac{1}{3}\).

Solution: Using the formula: $\(\tan \theta = \left|\frac{2 - (-\frac{1}{3})}{1 + 2(-\frac{1}{3})}\right|\)\( \)\(= \left|\frac{2 + \frac{1}{3}}{1 - \frac{2}{3}}\right|\)\( \)\(= \left|\frac{\frac{7}{3}}{\frac{1}{3}}\right| = 7\)$

Therefore, \(\theta = \arctan(7) \approx 81.87°\)

Example 2: Angle Between Lines in General Form Find the angle between \(2x + 3y - 5 = 0\) and \(3x - 2y + 7 = 0\).

Solution: First, find the slopes:

  • Line 1: \(m_1 = -\frac{2}{3}\)

  • Line 2: \(m_2 = \frac{3}{2}\)

Using the formula: $\(\tan \theta = \left|\frac{-\frac{2}{3} - \frac{3}{2}}{1 + (-\frac{2}{3})(\frac{3}{2})}\right|\)\( \)\(= \left|\frac{-\frac{4}{6} - \frac{9}{6}}{1 - 1}\right|\)$

Since the denominator is 0, the lines are perpendicular, so \(\theta = 90°\).

Example 3: Using Direction Vectors Find the angle between lines with direction vectors \(\vec{v_1} = (3, 4)\) and \(\vec{v_2} = (5, -12)\).

Solution: $\(\cos \theta = \frac{|3(5) + 4(-12)|}{\sqrt{3^2 + 4^2}\sqrt{5^2 + (-12)^2}}\)\( \)\(= \frac{|15 - 48|}{\sqrt{9 + 16}\sqrt{25 + 144}}\)\( \)\(= \frac{33}{5 \times 13} = \frac{33}{65}\)$

Therefore, \(\theta = \arccos\left(\frac{33}{65}\right) \approx 59.49°\)

Multiple Choice Questions#

Sector Specific Questions: Angle Between Lines Applications#

Key Takeaways#

Important

Angle Between Lines - Essential Concepts

  1. Slope Formula: For lines with slopes \(m_1\) and \(m_2\): $\(\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1m_2}\right|\)$

  2. Special Cases:

    • Parallel lines: \(m_1 = m_2\) → \(\theta = 0°\)

    • Perpendicular lines: \(m_1m_2 = -1\) → \(\theta = 90°\)

  3. Vector Method: For direction vectors \(\vec{v_1}\) and \(\vec{v_2}\): $\(\cos \theta = \frac{|\vec{v_1} \cdot \vec{v_2}|}{|\vec{v_1}||\vec{v_2}|}\)$

  4. Applications: Used in road design, crystallography, financial analysis, and architecture

  5. Remember: Always take the acute angle (0° to 90°) unless otherwise specified

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