Two Dimensions- Angle Between Two Lines

The angle between the two lines is defined as the measure of inclination, which tells the relationship between the two lines. When the two lines (straight) intersect each other, it creates two angles. When the lines are not at right angles, i.e., 90°, the intersection forms either an obtuse angle or an acute angle. So, at the point of intersection, there is usually a formation of two angles, the first is an acute angle, and the second is obtuse. Both the angles are supplementary to each other. When the sum of two angles is equal to 180°, it is known as supplementary angles. To understand Two Dimensions-Angle Between Two Lines, carefully read this article.

How Can One Find Two Dimensions-Angle Between Two Lines?

1. Find Angle That Is Created Between Two Lines

Consider the three coordinates present on the given x-axis as well as another three on the y-axis, the coordinates of which are already given to you.

Now, take a line of which the coordinates are  (x₁, y₁) and (x₂, y₂). 

Now, the equation that comes as the slope is: 

m(Slope) = y₂ – y₁/x₂ – x₁

m1 and m2 can be evaluated if you substitute the equation given above in the given formula above then the result of m2 and m1 can be added in the given formula that is:

tan θ = ± (m₁ – m₂ ) / (1- m₁ x m₂)

2. The second situation, if they are parallel lines:

Now, if the lines come to as parallel in nature, that denotes the angle created in between the two lines is at 0°

tan Ɵ =0

m₁ – m₂/1+m₁m₂=0

m₁ – m₂=0

m₁ = m₂

Hence, if the slopes come to be equal, the lines are considered to be parallel. 

Reminder: The result of the angle tan Ɵ would always remain to be positive. 

 3. The third situation is if Lines are considered to be Perpendicular.

So, if the lines are considered perpendicular to each other, the angle created between the two would be 90°. By this, it means the angle Ɵ is equal to 90° then,

1/tan Ɵ = 0

1+m₁m₂/m₁-m₂ = 0

1+m₁m₂ =0

m₁m₂ = -1

Now the multiplication result of the slope comes as -1, which advocates that the given lines are perpendicular to each other.

This is how you can find Two Dimensions-Angle Between Two Lines.

Two Dimensions-Angle Between Two Lines Formulae

Suppose two lines are non-parallel to each other. Each of them has a slope of m2 and m1, respectively. The angle that is created between them is Ɵ, then the value of the formula that the two lines make comes to as:  

tan Ɵ = (m₂ – m₁) / (1 + m₁m₂)

Deriving Formula of Two Dimensions-Angle Between Two Lines

Considering that, there is a two-dimensional plane with a y-axis and an x-axis.

There are two lines that are intersecting and creating an obtuse angle and an acute angle, respectively, 

Now, if we consider the angle that is acute as Ɵ 

Step 1 will follow up as 

θ = θ₂ – θ₁

To prove step 1 above, consider a triangle ABC, then,

By the property of angle sum, one can consider that: 

θ + θ₂ + x = 180 ……..(1)

x + θ₂ = 180…………. (2) 

(as the angle, i.e., x and θ2 frames a pair linear in nature, i.e., 180°)

The third step includes, from the above two equations, we can see that:

θ + θ₁ + x = x + θ₂ = 180 

Now, if we subtract x from each sides we receive:

θ + θ₁ + x – x = x + θ₂ – x 

 you will receive,

θ + θ₁ = θ₂

The fourth step is to subtract the angle θ₁ from each side, then.

θ + θ₁ – θ₁ = θ₂ – θ₁

you will receive,

θ = θ₂ – θ₁

Finally, if we apply the tangents result on both the ends: 

tan θ = tan ( θ₂ – θ₁)

Now, if we use the result of the formula of the tangent, you will receive:

tan θ = tan θ₂ – tan θ₁/1+tan θ₁tanθ₂

Then from the above formula of the line inclination, we can get the angle. 

tan θ = m

Therefore one can substitute the result i.e.,tan θ₂ = m₂ and tan θ₁ = m₁ we will receive,

tan Ɵ= |(m₂ – m₁)/(1+m₁m₂)

Two Dimensions- Angle Between Two Lines formula will quickly sum up your given problem.

The angle between Two Lines in 3D

In the 3D space, straight lines are represented in two forms: Cartesian form and vector form.

Given below is the method of Cartesian Form:

L1: (x – x1) / a1 = (y – y1) / b1 = (z – z1) / c1

L2: (x – x2) / a2 = (y – y2) / b2 = (z – z2) / c2

Here L1 & L2 represent the two straight lines that are passing through the points (x1, y1, z1) and (x2, y2, z2) respectively in three-dimensional space in Cartesian Form. 

Direction ratios of line L1 are a1, b1, c1then a vector parallel to L1 is 

L1 = a1 i + b1 j + c1k

Direction ratios of line L2 are a2, b2, c2, then a vector parallel to L2 is 

L2 = a2i + b2 j + c2k

Then the angle ∅ between L1 and L2 is given by:

∅ = cos-1{(1 . 2) / (|L1| × |L2|)}

Conclusion

So, the process of derivation and its formula will help you find Two Dimensions-Angle Between Two Lines. There are always alternatives to the solution, so it depends on your understanding of the problem and the best solution you can put into finding it. 

The article has explained how one can calculate the angle between two lines in a two and three-dimensional space. It will help you have proper knowledge on the subject.