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Angles between two lines

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The Angles between two lines is defined as shown below:
If the lines are parallel then the angles between both lines is zero.

Suppose we have angle ‘∅’in between two lines then it always satisfy the inequalities condition.
=>
0 $\leq$ ∅ $\leq$ $\frac{\pi}{2}$

If the Slope of both the lines is a1 and a2 then the angle ∅ is obtained from:

Then we can write it as:

=> Tan ∅= | $\frac{a_{1} - a_{2}}{1+a_{1} a_{2}}$ |

When we find the value of a1 a2 by above formula then we get:

=>1 + a1 a2 = 0;

=> a1 a2 = -1;

This condition follows when the lines are perpendicular and also ∅
equal to $\frac{\pi}{2}$.

Out of both lines, one line is parallel to y – axis then there is no Slope and the angle ∅ is deduced by using the other line.

Suppose we have one slope equals to zero then the angle between both the lines is the angle between the x – axis and one of the lines.

Let slope of one line is zero and slope of other line is ‘a’, then we get the formula which is given below:

=> Tan ∅ = |a|;

Suppose one of the slope is infinite and that line is parallel to the y – axis, then the angles between both the lines is same as the angle between one line (which has slope ‘a’) and y – axis.

So the formula for this statement is given below:

=> Tan ∅ = |1 / m|,

We get this formula when one of the slope a1 a2 approaches to ∞.

We find the angle between two lines by using vector product method by using formula which is given below:

Cos ∅ = | $\frac{\vec{m} . \vec{n}}{(\vec{m}) (\vec{n})}$ |

Where ‘m’ and ‘n’ are two vectors and ∅ is angle between two lines.

Angle between two line formula

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The Intersection between two lines defined as an Angle between two lines. Here we will see how to find the angle between two lines. The formula is given by:
⇒ tan∅ = tan (a2 – a1),
= (tan a2 - tan a1 )/ (1 + tan a1 tan a2 )
To find the angle between two lines we have to follow the steps explained below:
Step1: To find the angle first of all we have to assume the angle between two lines.
Step2: Then find the value of slopes of the given lines. Let the line be ‘A1’ and ‘A2’.
Step3: Now determine the Tangent difference between two angles using trigonometric Functions.
Step4: Using these steps we can easily get the angle between two lines.
We know that,
a2 = a1 + ∅,
∅ = a2 – a1,
Now, calculate the value of slopes of line ‘A1’ and ‘A2’.
Using trigonometric Functions:
tan∅ = tan (a2 – a1),
=(tan a2 - tan a1 )/ (1 + tan a1 tan a2 )
If tan a1 = m1 (m1 is the Slope of the line A1)
⇒ tan a2 = m2 (m2 is the Slope of the line A2)
So, value of ‘m2’ is greater than ‘m1’, (m2 > m1).
Substitute these values in the given equation and we get:
Tan ∅ = (m2 -m1)/ ( 1 + m1 m2 )
Using this formula we find angles between two lines.
Suppose the value of m1 = ½ and m2 = 4 t and we have to find the angle between two lines m1 and m2?
We know that the value of m2 is always equal to the value of m1.
Given, m1 = ½ and m2 = 4, the formula to find the angle between two lines is given as:
tan∅ = (m2 -m1)/ ( 1 + m1 m2 )
Plug in value of m1 and m2 in the given formula:
Tan ∅ = {4-(1/2)}/(1 + ½ * 4),
={(8-1)/2}/(1 + 2),
tan∅ = 7/6.
This is all about the angle between two line formula and how to use it while solving problems related to it.