Angle Between Two Lines
The angle between is the measure of the inclination between the two lines. For two intersecting lines, there are two types of angles between the lines, the acute angle and the obtuse angle. Here we consider the acute angle between the lines to be the angle between two lines.
The angle between two lines whose slopes are m1 and m2 is given by the formula tan-1|(m1 - m2)/(1 + m1 m2)|. Let us check all the formulas related to the angle between two lines in a coordinate plane and three-dimensional space.
| 1. | How to Find Angle Between Two Lines? |
| 2. | Formulas For Angle Between Two Lines |
| 3. | Angle Between Two Lines in Three Dimensional Space |
| 4. | FAQs on Angle Between Two Lines |
How to Find Angle Between Two Lines?
The angle between two lines can be calculated by knowing the slopes of the two lines, or by knowing the equations 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.
tan θ = \(\left|\dfrac{m_1 - m_2}{1 + m_1m_2}\right|\)
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.
tan θ =\(\left|\dfrac{a_2b_1 - a_1b_2}{a_1a_2 + b_1b_2}\right|\)
Formulas for Angle Between Two Lines
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 have equal slopes (\(m_1 = m_2\)) is 0º.
- The angle between two lines which are perpendicular to each other and have the product of their slopes equal to -1 (\(m_1m_2 = -1\)) is 90º.
- The angle between two lines having slopes \(m_1\), and \(m_2\) respectively is θ = \(tan^{-1}\left|\dfrac{m_1 - m_2}{1 + m_1m_2}\right|\).
- 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 =\(tan^{-1}\left|\frac{a_2b_1 - a_1b_2}{a_1a_2 + b_1b_2}\right|\)
- 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 cos θ = \(\frac{|a_1a_2 + b_1b_2| }{\sqrt {a_1^2 + b_1^2 } \cdot \sqrt{a_2^2 + b_2^2 }}\)
- The angle between a pair of straight lines ax2 + 2hxy + by2 = 0 is θ = \(Tan^{-1}\frac {2\sqrt {(h^2 - ab)}}{(a + b)}\)
- By the cosine rule, in a triangle having sides of lengths, a, b, c, the angle between two sides of a triangle is equal to cos A = \(\frac{b^2 + c^2 - a^2}{2bc}\).
Angle Between Two Lines In Three Dimensional Space
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.
cos θ = \(\dfrac{b_1\cdot b_2}{|b_1||b_2|}\)
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.
cos θ = \(\dfrac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}\)
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.
cos θ = \(|l_1l_2 + m_1m_2 + n_1n_2|\)
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FAQs on Angle Between Two Lines
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 tan θ = \(\left|\dfrac{m_1 - m_2}{1 + m_1m_2}\right|\).
What is the Angle Between Two Lines Formula?
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 tan θ = \(\left|\dfrac{m_1 - m_2}{1 + m_1m_2}\right|\).
- 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 tan θ =\(\left|\dfrac{a_2b_1 - a_1b_2}{a_1a_2 + b_1b_2}\right|\).
Can the Angle Between Two Lines be Negative?
No, the angle between the two lines is always positive. Also, the formula to find it always includes absolute value sign in it.
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 cos θ = \(\left|\dfrac{b_1 \cdot b_2}{|b_|.|b_2|}\right|\).
Why is the Absolute Value Sign in the Angle Between Two Lines Formula?
The angle between two lines refers to the acute angle between them. The tan θ or cos θ formula which we use to find the angle between lines leads to an obtuse angle if the absolute value is NOT included. That is why, we include it.