Angle Between Two Lines

Angle between two lines helps to know the relationship between the two lines. It is the measure of the inclination between the two lines. For two intersecting lines, there are two angles between the lines, the acute angle and the obtuse angle. Here we consider the acute angle between the lines, for the angle between two lines.

The angle between two lines is useful to find the measure of the angle between two sides of a closed polygon. Let us check the formulas and examples for the angle between two lines in a coordinate plane, and three-dimensional space..

The angle between two lines can be calculated by knowing the slope of the two lines, or by knowing the equation of the two lines. The angle between two lines generally gives the acute angle between the two lines.

The angle between two lines can be computed from the slope of the two lines, and by using the trigonometric tangent function. Let us consider two lines with slopes \(m_1\), and \(m_2\) respectively. The acute angle θ between the lines can be calculated using the formula of the tangent function. The acute angle between the two lines is given by the following formula.

Tanθ = \(\dfrac{m_1 - m_2}{1 + m_1.m_2}\)

Further, we can find the angle between the two lines if the equations of the two lines are given. Let the equations of the two lines be \(l_1 = a_1x + b_1y + c_1 = 0\), and \(l_2 = a_2x + b_2y + c_2 = 0\). The angle between the two lines can be computed by the tangent of the angle between the two lines.

Tanθ =\(\dfrac{a_2b_1 - a_1b_2}{a_1a_2 + b_1b_2}\)

The following different formulas help in easily finding the angle between two lines.

• The angle between two lines, of which, one of the line is ax + by + c = 0, and the other line is the x-axis, is θ = Tan^-1(-a/b).
• The angle between two lines, of which one of the line is y = mx + c and the other line is the x-axis, is θ = Tan^-1m.
• The angle between two lines that are parallel to each other and having equal slopes (\(m_1 = m_2\)) is 0º.
• The angle between two lines which are perpendicular to each other and having the product of their slopes equal to -1 (\(m_1.m_2 = -1\)) is 90º.
• The angle between two lines having slopes \(m_1\), and \(m_2\) respectively is θ = \(Tan^{-1}\dfrac{m_1 - m_2}{1 + m_1.m_2}\).
• The angle between two lines having equations \(l_1 = a_1x + b_1y + c_1 = 0\), and \(l_2 = a_2x + b_2y + c_2 = 0\) is θ =\(Tan^{-1}\frac{a_2b_1 - a_1b_2}{a_1a_2 + b_1b_2}\)
• The angle between two lines having equations \(l_1 = a_1x + b_1y + c_1 = 0\), and \(l_2 = a_2x + b_2y + c_2 = 0\) is Cosθ = \(\frac{a_1.a_2 + b_1.b_2 }{\sqrt {a_1^2 + b_1^2 }. \sqrt{a_2^2 + b_2^2 }}\)
• The angle between a pair of straight lines ax² + 2hxy + by² = 0 is θ = \(Tan^{-1}\frac {2\sqrt {(h^2 - ab)}}{(a + b)}\)
• In a triangle having sides of lengths, a, b, c, the angle between two sides of a triangle is equal to CosA = \(\frac{b^2 + c^2 - a^2}{2bc}\)

The angle between two lines in a three-dimensional space can be calculated similarly to the angle between the two lines in a coordinate plane. For two lines with equations \(r = a_1 + λb_1\), and \(r = a_2 + λb_2\), the angle between the lines is given by the following formula.

Cosθ = \(\dfrac{b_1.b_2}{|b_1|.|b_2|}\)

Further for two lines having the direction ratios as \((a_1, b_1, c_1)\), and \((a_2, b_2, c_2)\), the angle between the lines can be computed using the below formula.

Cosθ = \(\dfrac{a_1.a_2 + b_1.b_2 + c_1.c_2}{\sqrt {a_1^2 + b_1^2 + c_1^2}. \sqrt{a_2^2 + b_2^2 + c_2^2}}\)

Also for two lines having the direction cosines as \(l_1, m_1, n_1\), and \(l_2, m_2, n_2\), the angle between the two lines can be computed using the following formula.

Cosθ = \(|l_1.l_2 + m_1.m_2 + n_1.n_2|\)

Related Topics

The following topics help in clearly understanding the concept of the angle between two lines.

• Coordinate Geometry
• Equation of a Line
• Parallel Lines
• Types of Angles

How Do You Find the Angle Between Two Lines?

The angle between two lines can be calculated from the slopes of the lines or from the equation of the two lines. The simplest formula to find the angle between the two lines is from the slope of the two lines. The angle between two lines with slopes \(m_1\), and \(m_2\) respectively is Tanθ = \(\dfrac{m_1 - m_2}{1 + m_1.m_2}\).

What Is the Formula To Find Angle Between Two Lines?

The are two important formulas to find the angle between two lines in a coordinate plane. For two lines having slopes \(m_1\), and \(m_2\) respectively, the angle between the two lines is Tanθ = \(\dfrac{m_1 - m_2}{1 + m_1.m_2}\). And the second formula to find the angle between the two lines having the equations \(l_1 = a_1x + b_1y + c_1 = 0\), and \(l_2 = a_2x + b_2y + c_2 = 0\), is Tanθ =\(\dfrac{a_2b_1 - a_1b_2}{a_1a_2 + b_1b_2}\).

How To Find Angle Between Two Lines in Three Dimensional Geometry?

The angle between two lines in three dimensional geometry, having the equations of the lines as \(r = a_1 + λb_1\), and \(r = a_2 + λb_2\), is Cosθ = \(\dfrac{b_1.b_2}{|b_1|.|b_2|}\).