2d geometry2d geometrypoints, lines, vectorspoints, lines, vectors

2

Geometric primitives

• point• line• line segment• polygon

3

Coordinate system

• geometry versus algebra• point: pair of numbers• line: equation

Equation of line l?

4

Equation of a line

• slope: a• y-intercept: b• vertical lines: no representation

y=ax+b

5

Vector: geometrically

• directed line segment• magnitude: length• direction

AB

6

vector: algebraically

• pair of numbers• displacement in two directions

=

21

v

7

point versus vector

• point A = (-1,2)

• vector

• mathematically equivalent• conceptually different

−=

21

OA

point

vector

8

vector addition

++

db

ca =

d

c +

b

a

34

= 21

+ 13

9

vector inversion

−−

−

ba

= ba

10

vector subtraction

−−

=

−−

+

−

tysx

ts

yx

= ts

yx

11

pointToPoint vector

a - b AB

b a- AB

OB AO AB

=

⇒+=

⇒+=

x

y

O

a

b

A (a1,a2)

B (b1,b2)

12

scalar multiplication

=

=

a

a

a

aa

2

1

2

1

λ

λλλ

=

6

3

2

13

13

vector equation of a line

=

r

r

s

s

yx

2

1

2

1 + λx = s + λr

x1 = s + rx2 = s + 2rx3 = s – rx4 = s + ½r

14

direction and place

direction Vector: r

place Vector: sx = s + λr

15

parametric equations

rλs y

rλ +sx

22

11

+=

=

=

r

r λ +

s

s

y

x

2

1

2

1

+=

⇔

rλ +s

rλs

y

x

22

11

parameter: λ

16

line through two points

line AB: x = a + λ(b - a)

a - b ABr ==

a OAs ==

17

Exercise

• Given two points A=(-1,2) en B=(3,-1)• Give the (normal) equation of line l

through A and B• Give the vector equation of line l

18

equation to vectors

=

25

2-0

yx

λ +

21052 52 −=⇔=− xyyx

22

5

−==

λλ

y

x

ThenChoose

vector equation:

equation:

19

vectors to equation

−=

1

4

3

1

y

x λ + vector equation:

parametric equations: λ

λ +

+=

=

3y4-1x

λ

λ +

+=

=

4124y4-1x

eliminate λ: 134 −=− yx⇒

what do you notice when you compare direction vector and equation?

20

intersection of two lines

two equations: yx

yx

−=−

=−

53

623basic skills

−=

14

31

yx

λ + two vector equations:

−

−

=

32

111

yx

μ +

different parameters!

hard work

21

intersection (2)

−

=

12

16

yx

λ +

33 =− yx

vector equation line m:

equation line l:

y

x

yx

−=+=

=−

λλ

1

26

33

combine equations:

3 equations and 3 unknowns

22

substitution method

y

x

yx

−=+=

=−

λλ

1

26

33substitute x and y

3)1()26(3 =−−+ λλ

33 =− yx

gives

solve for λ:

2

147

31618

−=⇔−=

⇔=+−+

λλ

λλ

23

solution

y

x

yx

−=+=

=−

λλ

1

26

33 find x and y:( ) 2226 =−⋅+=x

( ) 321 =−−=y2−=λ

intersection (2,3)

24

exercise

• reader exercise 2.5.a

=

1-

4 +

2

1

y

xλ

line m: 3x + 2y = -3

line l: