Angle Between Two Lines

Let us learn the basics of two-dimensional geometry with this lesson. What is a straight line? How is the angle between two lines intersecting straight lines measured? Understand the concepts, calculations and how to solve problems with the help of a solved example in this lesson.

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Straight Lines in Geometry

In mathematics, straight lines have an important role to play in two-dimensional geometry. A straight line is nothing but a locus of all such infinite number of points lying in the two-dimensional space and extending out in either direction infinitely. Thus, a straight line (also referred to as a ‘line’) has no height but only, length.

Browse more Topics under Three Dimensional Geometry

• Angle Between a Line and a Plane
• Coplanarity of Two Lines
• Angle Between Two Planes
• Direction Cosines and Direction Ratios of a Line
• Distance Between Parallel Lines
• The Distance Between Two Skew Lines
• Distance of a Point from a Plane
• Equation of a Plane in Normal Form
• Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
• The Equation of Line for Space
• Equation of Plane Passing Through Three Non Collinear Points
• Intercept Form of the Equation of a Plane
• Plane Passing Through the Intersection of Two Given Planes

The Angle Between Two Lines

In the following section, we shall now move on to understand how the angle between two straight lines is calculated. Note that when we refer to the angle between two lines, in normal cases, we are actually referring to the angle between two intersecting lines. This is because the angle between two perpendicular lines is 90º (by definition) and that between two parallel lines will be 0º.

Hence, we will now look at how the angle between two intersecting lines is calculated. The following figure will help you understand the geometric implication of this calculation better.

Source: MathWorks

Let the direction cosines of the two lines are (l₁, m₁, n₁) and (l₂, m₂, n₂) respectively. Recall that the direction cosines of a line are actually the angles between the line and either of the three coordinate axes. Now, let ɵ be the angle between the lines. Then, note the formula:

Cos Θ = | l₁l₂ + m₁m₂ + n₁n₂| /  l₁² + m₁² + n₁² )^1/2 (l₂² + m₂² + n₂²)^1/2

If you wish to find the angle in terms of Sin Θ, you can easily do so using the formula:

Sin ²Θ= 1 – Cos² Θ, and then you can replace Cos Θ using the formula above.

An important point to note here is that we have considered the lines to be passing through the origin. A solved problem will help you understand this better.

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Solved Example for You

Question 1: Find the angle between the pair of lines given by:

(x + 3)/3 = (y – 1)/5 = (z + 3)/4 and (x + 1)/1 = (y – 4)/1 = (z – 5)/2

Answer: We can solve this problem by finding the cosine of the angle between the two lines and then taking an inverse of the cosine. Let Θ be the line between the two lines. We know from the formula that:

Cos Θ = (3.1 + 5.1 + 4.2) / ( 3² + 5² + 4² ) ^1/2 (1² + 1² + 1²) ^1/2

Cos Θ = 16/ 50^1/2 . 6^1/2

Cos Θ = 16/ 10. 3^1/2

Hence, Θ = Cos ^-1 (16/ 10. 3^1/2 ) is the required angle.

Question 2: Explain the way of finding the angle between the two lines with regards to a triangle?

Answer: First of all, one must make use of the cosine rule. Afterwards, one must plug in the values for the sides b, c, and the angle A. After that, one must solve for side a. Then the angle value must be used along with the sine rule to deal with angle B. Finally, one can find angle C by using the knowledge that the angles of all triangles add up to 180 degrees.

Question 3: Explain the angle between two perpendicular lines?

Answer: The angle between two perpendicular lines always happens to be 90°. This is called as the right angle. The characteristic of being perpendicular refers to the relationship that exists between two lines whose meeting takes place at a right angle (90 degrees).

Question 4: Explain the angle between the two vectors?

Answer: The angle between the two vectors can be defined as an angle between two sides of a two-dimensional triangle whose lengths are ||a||, ||b||, and ||a-b||.

Question 5: What is the name of the 360 degree angle?

Answer: The 360 degree angle is called a complete angle.