Angle Between Straight Lines

[MathDefinition title="Angle Between Two Lines"] The angle between two lines is the smaller angle formed at their point of intersection. It depends on the slopes of the lines. [/MathDefinition]

[MathDefinition title="Angle from Slopes"] If m1 and m2 are the slopes of two lines, then: tan θ = |(m2 - m1)/(1 + m1*m2)| where θ is the acute angle between the lines. [/MathDefinition]

[MathKeyFormula title="Key Formulas"]

1. tan θ = |(m2 - m1)/(1 + m1*m2)|
2. For perpendicular lines: m1 * m2 = -1
3. For parallel lines: m1 = m2
4. Angle with x-axis: tan θ = m (slope)
5. If lines are a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0: cos θ = |a1a2 + b1b2|/√((a1^2+b1^2)(a2^2+b2^2)) [/MathKeyFormula]

[MathNote title="Special Cases"]

• Perpendicular lines: Product of slopes = -1
• Parallel lines: Slopes are equal
• If one line is vertical (undefined slope), angle = 90° - arctan(m) [/MathNote]

[MathExample title="Example 1: Finding Angle Between Lines"] Problem: Find the angle between the lines y = 2x + 3 and y = -x + 1.

Solution: Slope of first line (m1) = 2 Slope of second line (m2) = -1

Using tan θ = |(m2 - m1)/(1 + m1m2)|: tan θ = |(-1 - 2)/(1 + 2(-1))| = |-3/(1 - 2)| = |-3/(-1)| = |3| = 3

θ = tan^(-1)(3) ≈ 71.57°

Therefore, the angle between the lines is approximately 71.57°. [/MathExample]

[MathExample title="Example 2: Perpendicular Lines"] Problem: Find the value of k if the lines 2x + 3y = 7 and kx - 4y = 5 are perpendicular.

Solution: For perpendicular lines: m1 * m2 = -1

First line: 2x + 3y = 7 → y = (-2/3)x + 7/3, so m1 = -2/3

Second line: kx - 4y = 5 → y = (k/4)x - 5/4, so m2 = k/4

m1 * m2 = -1 (-2/3) * (k/4) = -1 -k/6 = -1 k = 6

Therefore, k = 6. [/MathExample]

[MathExample title="Example 3: Acute and Obtuse Angles"] Problem: Find both the acute and obtuse angles between the lines y = x and y = √3x.

Solution: Slope of first line (m1) = 1 Slope of second line (m2) = √3

tan θ = |(√3 - 1)/(1 + √3)| = |(√3 - 1)/(1 + √3)| * (√3 - 1)/(√3 - 1) = |(√3 - 1)^2/(3 - 1)| = |(3 - 2√3 + 1)/2| = |(4 - 2√3)/2| = |2 - √3| = 2 - √3 (since 2 > √3)

Acute angle θ = tan^(-1)(2 - √3) = 15°

Obtuse angle = 180° - 15° = 165°

Therefore, the acute angle is 15° and the obtuse angle is 165°. [/MathExample]

[MathExample title="Example 4: Angle with x-axis"] Problem: Find the angle that the line 3x - 4y + 10 = 0 makes with the positive direction of the x-axis.

Solution: Rewrite in slope-intercept form: 3x - 4y + 10 = 0 -4y = -3x - 10 y = (3/4)x + 5/2

Slope m = 3/4

tan θ = m = 3/4

θ = tan^(-1)(3/4) ≈ 36.87°

Therefore, the line makes an angle of approximately 36.87° with the positive x-axis. [/MathExample]