sim tec th1 vector basics
TRANSCRIPT
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2d geometry2d geometrypoints, lines, vectorspoints, lines, vectors
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Geometric primitives
• point• line• line segment• polygon
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Coordinate system
• geometry versus algebra• point: pair of numbers• line: equation
Equation of line l?
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Equation of a line
• slope: a• y-intercept: b• vertical lines: no representation
y=ax+b
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Vector: geometrically
• directed line segment• magnitude: length• direction
AB
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vector: algebraically
• pair of numbers• displacement in two directions
=
21
v
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point versus vector
• point A = (-1,2)
• vector
• mathematically equivalent• conceptually different
−=
21
OA
point
vector
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vector addition
++
db
ca =
d
c +
b
a
34
= 21
+ 13
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vector inversion
−−
−
ba
= ba
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vector subtraction
−−
=
−−
+
−
tysx
ts
yx
= ts
yx
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pointToPoint vector
a - b AB
b a- AB
OB AO AB
=
⇒+=
⇒+=
x
y
O
a
b
A (a1,a2)
B (b1,b2)
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scalar multiplication
=
=
a
a
a
aa
2
1
2
1
λ
λλλ
=
6
3
2
13
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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
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direction and place
direction Vector: r
place Vector: sx = s + λr
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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
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line through two points
line AB: x = a + λ(b - a)
a - b ABr ==
a OAs ==
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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
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equation to vectors
=
25
2-0
yx
λ +
21052 52 −=⇔=− xyyx
22
5
−==
λλ
y
x
ThenChoose
vector equation:
equation:
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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?
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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
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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
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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
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solution
y
x
yx
−=+=
=−
λλ
1
26
33 find x and y:( ) 2226 =−⋅+=x
( ) 321 =−−=y2−=λ
intersection (2,3)
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exercise
• reader exercise 2.5.a
=
1-
4 +
2
1
y
xλ
line m: 3x + 2y = -3
line l: