4. Application of vectors to metric problems: THE ANGLE BETWEEN TWO LINES
Block:Geometry
 
Application of vectors to metric problems
4.2. ANGLE BETWEEN TWO LINES
The smaller of the angles formed is called the angle between two lines. It can be found in two forms:
First form: The angle, A, between two lines r1 and r2, can be obtained from their direction vectors,

d1, d2, or from their normal vectors, n1, n2:

Second form: If the lines are given in implicit form: r1:Ax+By+C = 0  and r2:A'x+B'y+ C' = 0

To find the angle between r1 and r2, we can take the normal vectors n1(A,B) and n2(A',B') 

First form:

1.-Observe in the figure that we have two lines, r1 and r2, in parametric form, the simplest way to find the angle a which is formed is, taking the direction vectors, which are already given in the equations. 

Move the furthest point of the direction vector of r1, or that is d1, or change its coordinates using the buttons at the bottom. In this way the direction of r1 is changed and therefore the angle of r1 with r2.

2.-Observe that when the angle between the vectors is greater than 90º, the supplement is taken for the angle between the lines, as this will always be the minor angle between the two. 

3.- Use your calculator to work out the answers and verify them in the figure.

4.- Copy down the the equations of the lines given in the previous figure, your direction vectors , the scalar products of them, the modulus of them, and substitute everything into the formula which gives us the cosine of the angle formed by the two lines. Then copy down the result from the figure, check it with the calculator.

5.- Calculate the angle formed by the lines:    and 
Verify the result in the previous figure.

Second form:

6.-Copy into your workbook the equations of the lines given in the previous figure, their normal vectors, their scalar products, their modulos, and substitute everything into the formula which gives us the cosine of the angle formed by the two lines. Then copy the result from the figure, checking it with the calculator.

7.- Calculate the angle formed by the two lines:  r1:4x + 2y + 14 = 0  and  r2:x - 2y - 4 = 0
Check the result in the figure.

       
           
  Ángela Núñez Castaín
 
Ministry of Education , Social Afairs and Sport. Year 2001
 
 

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