HOW TO FIND THE ANGLE BETWEEN TWO LINES

HOW TO FIND THE ANGLE BETWEEN TWO LINES WHEN THE EQUATIONS OF GIVEN LINES ARE  IN CARTESIAN  FORMS

In this post we shall study How to find the angle between two lines ,angle between two lines vectors in Cartesian form, angle between two lines in 3d, angle between two lines calculator, angle between two lines coordinate geometry,derivation of angle between two lines

Problem 1

Consider two lines whose equations are given in cartesian form as

Note that direction Ratios of 1st line is (1 , 2 , 3) and the direction Ratios of 2nd line is (2 , 3 , 4).

Note that Direction Ratios of  any line are those numbers written is the denominator in standard form of the equation of the line

we know that if a_1,b_1,c_1   and    a_2,b_2,c_{2,  are Direction Ratios of line} L_{1  and Line} L<sub>2 .

If  θ be the angle between two lines ,Then this angle  can be formulated as follows</sub>

So putting the values of direction ratios of both the lines in (3) ;

we  get 

cos θ = [2 + 6 + 12]/sqrt[ 1+ 4+ 9]×sqrt[ 4 +9 +16 ]

cos θ = [20]/√[ 14×29 ];

cos θ = 10/√203

∴ θ = cos^-1 (10/√203)

Hence  cos^-1 (10/√203) is the angle between two lines

How to find angle between two  line in vector and cartesian   forms

 

Problem 2

Let us consider these two lines in cartesian form

Since these equations of lines are not in standard form  , in order to reduce these equations to standard form we shall have to make the coeffs of x, y , z unity.

In 1st part of equation (1) divide the num and den by 2 , and in 2nd part divide num and den by 3. 

Similarly divide 2nd part of  equation (2) by 5. we can rewrite these equations as follows

After cancellation and simplification , we get equations of lines in standard form

Here Direction Ratios of 1st Line are  (2,1 ,3)  and Direction Ratios of 2nd Line are (1 ,2 , 4) so using the formula

Putting the values of direction ratios of both the lines in above formula ,we get

cos θ = [2 + 2 + 12]/sqrt[ 4+ 1+ 9]×sqrt[ 1 +4 +16 ]

cos θ = [16]/sqrt[14×21];

cos θ = 16/√294

∴ θ = cos^-1 (16/√294)

Hence  cos^-1 (16/√294) is the angle between two lines

Problem 3

How to prove that the given lines are parallel to each others, 

To test whether given lines are parallel to each other , just check their direction ratios , id they are proportional to each others , then The given lines would be parallel to each other.

If the Direction ratios of 1st line is (1,2,5) and Direction ratios of 2nd  line is (2,4,10)  . If we take 2 common from direction ratios of 2nd lines then we have same direction Ratios as that of 1st line hence These lines are parallel o each other.

Problem 4

How to prove that the given lines are Perpendicular  to each others. If we calculate the angle between any two lines and it comes out to be 0 ( ZERO ) then  these would be perpendicular to each others.

E.g  If the D. R. of 1st line are ( 3, -1 , 3 )

And     D. R. of 2nd Line are  ( -2 ,3 ,3 ).

The cosine of angle between these  two line is zero

cos θ = 0

Then  θ = 90° 

⇒ Lines are Perpendicular to each others

Also Read previous posts 
How to Find perpendicular distance between skew lines

How to find slope of line ax+by=c


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To find the angle between two lines

x = y = z and

x = y = -z 


Want to check the solution of this problem ?

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Thanks for your  precious time to  read this post regarding how to find the angle between two lines in Cartesian form.