CARTESIAN EQUATIONS II
(LINES in R2)
A.
PARALLEL and PERPENDICULAR LINES
Two lines are parallel if:
𝑛1 = 𝑘𝑛2 , 𝑘 𝜖 𝑅, 𝑘 ≠ 0
Two lines are perpendicular if:
𝑛1 ∙ 𝑛2 = 0
Ex 
Ex 
Determine the C.E. of each of the following lines:
a)
passing through the point A(3,–2) and parallel to x + 6y – 2 = 0;
b)
with y–intercept of –4 and perpendicular to 𝑟 = −4,2 + 𝑡 5,3 , 𝑡 𝜀 𝑅.
Show that the given lines are parallel and non–coincident:
:
2:
1
3x – 4y – 6 = 0
6x – 8y + 12 = 0
B.
ANGLE BETWEEN TWO LINES in R2
The angle between two lines is the acute angle between their direction vectors.
1
2
𝑚1

Ex 
𝑚2

𝑐𝑜𝑠𝜃 =
𝑚1 ∙ 𝑚2
𝑚1 𝑚2
Determine the acute angle between the given lines:
:
2:
1
𝑟 = 1,7 + 𝑡 1,4 , 𝑡 𝜖 𝑅
𝑟 = 2,3 + 𝑡 3, −1 , 𝑡 𝜖 𝑅
The angle between two lines is the acute angle between their normal vectors.
1
𝑛2
2

𝑐𝑜𝑠𝜃 =
𝑛1 ∙ 𝑛2
𝑛1 𝑛2
𝑛1
Ex 
Determine the acute angle between the given lines:
:
2:
1
3x + 4y – 8 = 0
x – 2y + 6 = 0
HOMEWORK: p.443–444 #4, 5, 8, 10, 11, 14