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When two non-parallel straight lines intersect each other in a plane they form two opposite vertical angles. One among them is acute that is less than 90 degrees and the second one is obtuse which is more than 90 degrees. In simple words whenever two straight lines are intersecting each other they form two sets of angles, the line which forms an intersection creates a pair of acute and another pair of obtuse angles. The slopes of the intersecting lines form the absolute values of the angles.
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Key Terms: Angle between Two Lines, Angle between Two Lines Formula, Plane, Angles, Straight Lines, Parallel Lines
Formula for Angle between Two Straight Lines
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Taking into consideration the two non-parallel lines have slopes m1 and m2 and the angle between the lines is θ then the formula for finding the angle between the two lines will be:
tanθ = ±(m2-m1) / (1+m1m2)
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|---|---|
| Onto Function | Scalar matrix |
| Identity matrix | Rolle’s Theorem |
Derivation of the Formula for Two Straight Lines
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- It is to be noted that when two lines are intersecting each other one of the two pairs is acute and the other pair is obtuse.
- The acute angle denoted by a theta is the angle between the two lines which is defined as the smallest of these angles.
- The inclination of the two lines is to be used to find the angle between the two lines.
- Secondly its required to show that : θ = θ2 - θ1
- Looking at triangle ABC it is clear that the sum of the angles in the triangle is equal to 180 degrees.
- So the first equation will be :- θ + θ1 + x = 180
- Moreover, x + θ2 = 180 is the second equation due to the fact that x and θ2 are forming a straight line.
- After replacing 180 with x + θ2 in equation 1 we get θ + θ1 + x = x + θ2
- Now we subtract x from both sides
- θ + θ1 + x - x = x - x + θ2
- I.e, θ + θ1 = θ2 Is the solution
- Now we subtract θ1 from both sides
- θ + θ1 - θ1 = θ2 - θ1
- I.e, θ = θ2 - θ1 Is the solution
- Now we can use the formula for the tangent of the difference of two angles which is:
- tan θ = tan ( θ2 - θ1)
- tan θ = tan θ2 - tan θ1 / 1+ tan θ1 tan θ2
- Let tan θ be m
- Thus, tan θ1 = m1 and tan θ2 = m2
- After switching m1 for θ1 and m2 for θ2 in the above equation the final formula stands at:
- tan θ = +- (m1-m2) / (1+m1m2)
Read More: Area of a Triangle
Derivation of the Formula if the Lines are Perpendicular
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When two lines are perpendicular the angle between them is 90 degrees i.e,θ = 90°
1st step we will calculate
-1 is the product of the slope which shows that the lines are perpendicular to each other.
Read More: Properties of Determinants
Derivation of the Formula if the Lines are Parallel
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When two lines are parallel the angle between them is zero that is, θ = 0°
Now,
- m1 - m2 / 1 + m1m2 = 0
- m1 - m2= 0
- m1= m2
The slopes are equal to each other which shows that the lines are parallel.
Read More: Minors and Cofactors
How to Calculate Angle Between Two Lines in Coordinate Geometry?
Given, Coordinates of Three points on the x and y-axis. To consider,
Endpoints of a line having coordinates
- (x1 y1) (x2 y2)
The Equation for the slope,
- m = y2 - y1 / x2 - x1
Exchanging the values of m1 and m2 in the formula given below:
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| Related Topics | |
|---|---|
| Signum Function | Derivative of Inverse Trignometric Functions |
| Negative of a Vector | Co Planar Vectors |
Things to Remember
Sample Questions
Ques. How to determine the angle between two lines? (2 marks)
Ans. In an easy way one can determine the angles between two lines by placing a Protractor on one of the lines in a point where both the lines intersect each other, the other line points towards the degrees marked on the Protractor, to find out what degree it checks where the mark of other line is going.
Ques. What will be the angle between two lines x = 1 and y = x? (3 marks)
Ans. A the easiest way to find the angle between two lines is 2 find the slope of both the lines and then finding the angle that they make with x-axis using the following equation-
Tan A= m,
x=1…….(1)
y-x=0……(2)
Thus for (1)
If compared with a basic equation of line y=m×x+c
We obtain,
m = -1
For Equation (2)
We obtain,
m = -1
As, The slope of both the lines are equal, The angle between the lines is either 0 or 180 degree.
Ques. What are some daily life examples of an angle between two lines? (1 mark)
Ans. Some basic examples could be- hands of the clock, the fan blades, a pair of scissors etc.
Ques. How to find angles between two lines with regards to a triangle? (3 marks)
Ans. To find the angles between two lines with regards to a triangle one needs to make use of the cosine rule. Then the values of b,c and angle A need to be added in, then one must find the angle A and then the angle value must be used along with the sine rule to find angle B. To find angle C one must know that angles of all triangles if added sums up to 180 degrees this trick should be used to find Angle C.
Ques. What do you call a 360° angle?
Ans. A 360° angle is known as complete angle.
Ques. What is the angle between two perpendicular lines? (2 marks)
Ans. A perpendicular line is characterized by the relationship that exists between two lines which meet at a right angle. Thus, the angle between two perpendicular lines will always be 90° which is called the right angle.
Ques. What is the angle between two vectors? (1 mark)
Ans. The angle between two sides of two dimensional triangle which have lengths of ||a||, ||b||, and ||a-b|| are known as The angle Between two Vectors.
Ques. Find the equation of the line passing through (2, -1, 2) and (5, 3, 4) and of the plane passing through (2, 0, 3), (1, 1, 5), and (3, 2, 4). Also, find their point of intersection. (5 marks)
Ans.
Ques. If the lines (x−1)/−3 = (y−2)/2λ = (z−3)/2 and (x−1)/3λ = (y−1)/2 = (z−6)/−5 are perpendicular, find the value of λ. Hence find whether the lines are intersecting or not. (5 marks)
Ans.
Ques. Find the value of λ, so that the line (1−x)/3 = (7y−14)/λ = (z−3)/ 2 and (7−7x)/3λ = (y−5)/1 = (6−z)/5 are at right angles. Also, find whether the lines are intersecting or not. (5 marks)
Ans.
Ques. If 
respectively are the position vectors of points A, B, C and D, then find the angles between the straight lines 
are collinear or not. (5 marks)
Ans.
Ques. If a line makes angles 90° and 60° respectively with the positive directions of X and Y-axes, find the angle which it makes with the positive direction of the Z-axis. (5 marks)
Ans.
We know: I² + m² + n² = 1…(i)
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