Page 207 -

Solution: Choose two arbitrary points on this line. Denote their coordinates
by (x , y ) and (x , y ); being on the line, they satisfy the equation of the line:
1
1
2
2
ax + by + = 0
c
1 1
ax + by + = 0
c
2 2
Substracting the first equation from the second equation, we obtain:
ax( − x ) + b y( − y ) = 0
2 1 2 1
which means that (, ) (ab ⊥ x − x , y − y ), and the unit vector perpendicular
2 1 2 1
to the line is:

 a b 
ê =  , 
⊥
 a + b 2 a + b 
2
2
2
Example 7.5
Find the angle that the lines 3x + 2y + 2 = 0 and 2x – y + 1 = 0 and make
together.

Solution: The angle between two lines is equal to the angle between their nor-
mal unit vectors. The unit vectors normal to each of the lines are, respectively:

 3 2   2 − 1 
ˆ n =  ,  and ˆ n =  , 
1  13 13  2  5 5 

Having the two orthogonal unit vectors, it is a simple matter to compute the
angle between them:

ˆ ⋅
cos( )θ = nn ˆ = 4 ⇒ θ = . 1 0517 radians
1 2
65

7.2.1 MATLAB Representation of the Dot Product
The dot product is written as the product of a row vector by a column vector
of the same length.

Example 7.6
Find the dot product of the vectors:

© 2001 by CRC Press LLC