Learn How to Calculate Angle Between Two Lines - Tutorial

How to Calculate Angle Between Two Lines - Tutorial with Definition, Formula Example

Definition:

Angle between two lines is an angle formed when two straight lines intersect each other.

Formula:

angle-between-lines.png
Where
u1, u2, u3 = coordinates of U vector v1, v2, v3 = coordinates of V vector
Example 1 :

Calculate the angle between the following lines where the values are represented in Cartesian Form (x + 1) / 2 = (y + 1) / 3 = (z - 4) / 6 => u1 = 2, u2 = 3, u3 = 6 (x + 1) = (y + 2) / 2 = (z - 4) / 2 => v1 = 1, v2 = 2, v3 = 2

Given

In the given values the vector values are retrieved as follows: (u1,u2,u3) = (2,3,6) (v1,v2,v3) = (1,2,2)

Solution

Substitute the vector values in the formula,

cos α = |((2 x 1) + (3 x 2) + (6 x 2))| / 22 + 32 + 62 . 12 + 22 + 22
= |(2 + 6 + 12)| / 49 . 9
= |20| / (7 x 3)
= 20 / 21
α = cos-1 (0.9524)
α = 17.75o

Therefore, the angle between 2 lines is 17.75

Example 2:

Calculate the angle between the following lines where the values are represented in Equation Form r = 2x + 3y - z = 0, x - y + 2z = 0 s = 3x - y + z = 0, 2x + y - 3z = 0

Solution :
Step 1 : Calculation of u vector from first set of equation

u = i [(3 x 2) - (-1 x -1)] - j [(2 x 2) - (-1 x 1)] + k [(2 x -1) - (3 x 1)] u = i(6 - 1) - j(4 + 1) + k(-2 - 3) u = 5i - 5j - 5k u = (5,-5,-5)

Step 2 : Calculation of v vector from second set of equation

v = i [(-1 x -3) - (1 x 1)] - j [(-3 x 3) - (2 x 1)] + k [(3 x 1) - (2 x -1)] v = i(3 - 1) - j(-9 - 2) + k(3 + 2) v = 2i + 11j + 5k v = (2,11,5)

Step 3: Substituting vector values in formula
cos α = |((5 x 2) + (-5 x 11) + (-5 x 5))| / 52 + (-5)2 + (-5)2 . 22 + 112 + 52
= |10 - 55 - 25| / 75 . 150
= |-70| / 106.066
= 70 / 106.066
= 0.6599
α = cos-1(0.6599)
α = 48.7o

Therefore, the angle between 2 lines is 48.7

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