1. VECTOR
2006. 9.
류승택
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Vectors
• Super number
– Made up of two or more normal numbers, called
components
– Vector  a super number is associated with a distance
and direction
• Vector(벡터)
– Direct descendants of complex numbers
• Complex number(복소수) : a + b i (i = sqrt(–i) )
– A special class of numbers called hypercomplex numbers
(= hypernumbers)
– Vector properties
• Length and Direction
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Hypernumber
• Hypernumbers
– generalization of complex numbers
– Order is important
– N-dimension hypernumber
• (a1, a2, … , an), where a1, a2, …, an are the component of A
– Equality
• Must have same dimension
• A=B
– a1=b1, a2=b2, …, an=bn
– Addition | Subtraction
– Scalar multiplication
– Multiplication
• Not commutative(교환법칙)
– AB != BA
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Geometric Interpretation
• Geometric properties
– Displacement
•
•
•
•
First number : east if plus(+), or west if minus(-)
Second number: north (+) or south (-)
3D인경우: Third number: up(+) or down(-)
Ex) (16.3, -10.2) : 16.3(east), -10.2(south)
– Distance
• Pythagorean theorem
• Ex) (16.3) 2  (10.2) 2  19.23
– Direction
• Ex)
90  arctan
10.2
 122SE
16.3
result = tan theta
theta = arctan result
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Vector
• Vector
– W. R. Hamilton(1805-1865) : from the Latin word vectus (=
to carry over)
– input : an ordered pair of real numbers
– output : two real numbers that we interpret as magnitude
and direction
– 벡터의 표현
• 벡터는 진한 소문자로 표시
• 벡터의 성분은 컴마(,) 없이 각괄호( [ ) 로 표현
• Ex) a = [ a1 a2 ]
– Visualization
•
•
•
•
Distance-and-direction interpretation
Directed line segment or arrow
The length of the arrow: the magnitude of the vector
The orientation of the arrowhead : its direction
a
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Vector
• Vector addition
– A head-to-tail chain
– 2D: Parallelogram law (평행사변형의 법칙)
– 3D: rectangular parallelepiped(평행육면체의 법칙)
• The components must not be coplanar
• Free vectors
– No constrained vectors to any particular location
• Fixed (=bound) vectors
– Begin at a common point, usually the origin(원점) of a
coordinate system
• Distinction between free and fixed vectors
– Important for visualization and intuition
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Vector Properties
• Special vectors
– i, j, k, each of which has a length equal to one
•
•
•
•
i = [1 0 0]
j = [0 1 0]
k = [0 0 1]
Ex) a = [ax ay az]  a = ax + ay + az = axi+ ayj + azk
• Reverse the direction of any vector
– multiplying each component by -1
• Magnitude
– Positive scalar
– Ex) a  ax2  a y2  az2
• Unit vector
– Any vector whose length is equal to one
a
– Ex)
a y az 
a
ˆa   aˆ x aˆ y aˆ z   x
  cos 
a
a
a
a




Direction cosine of a
cos 
cos  
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Scalar Multiplication
• Multiplying any vector a by a scalar k  ka

– Ex) ka  ka x
ka y
ka z

• Magnitude ka
ka  ka 2x  ka 2y  ka 2z
ka  k a
• Possible effect of a scalar multiplier k
–
–
–
–
–
–
–
k > 1  Increase length
k = 1  No change
0 < k < 1  Decrease length
k = 0  Null vector ( 0 length)
-1 < k < 0  Decrease length and reverse direction
k = -1  Reverse direction only
k < -1  Increase length and reverse direction
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Vector Addition
• Given a = [ax ay az] and b = [bx by bz]]
– a + b = [ax+bx
ay+by
az+bz]
• Vector addition and scalar multiplication properties
–
–
–
–
–
a + b = b + a (교환법칙: commutative)
a + (b + c) = (a + b) + c (결합법칙: associative)
k(la) = kla
(k+l) a = ka +la
k(a + b)= ka + kb (배분법칙: distributive)
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Scalar and Vector Products
• Multiply two vectors  Two different ways
– Scalar product
• Produce a single real number
– Vector product
• Produce a vector
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Scalar Product
• Scalar Product (= dot product) (내적)
– The sum of the products of their corresponding components
a b  a x bx  a y by  a z bz
– Using the law of cosine, the angle between two vectors a and b
satisfies the equation
a b
a  b  a b cos
  cos 1
a  (b  c)  a  b  a  c
(ka)  b  a  (kb)  k (a  b)
ab

– Scalar Product Properties
ab  ba
2
a a  a
a  b  a b cos
a
b
Scalar ??
A quantity that is completely specified by its
magnitude and has no direction.
• If a is perpendicular to b, then a b  0
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Scalar Product
• Scalar Product
– Use the dot product to project a vector onto another
vector
VW
| X || W | cos | W |
 VW
| V || W |
• V  unit vector
• The dot product of V and W  the length the projection of W onto
V
– A property of dot product used in CG
• Sign
VW  0
if   90 o
VW  0
if   90
VW  0
if   90 o
V (unit vector)
W
X
o
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Scalar Product (참조)
• 풀이과정 a  b  a b cos
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Vector Product
• Vector Product (= Cross Product) 외적
a  b  (a y bz  a z by )i  (a xbz  a z bxz ) j  (a xby  a y bx )k

a  b  (a y bz  a z by )  (a xbz  a z bxz ) (a xby  a y bx )
i
j
k
a  b  ax
ay
az
bx
by
bz

a

b
– c = a x b  c is perpendicular to both a and b
• Perpendicular to the pane defined by a and b
• a sc = 0 ?? b sc = 0 ??
a c  a x (a y bz  az by )  a y (a xbz  a z bxz )  a z (a xby  a y bx )
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Vector Product
– If two vectors a and b parallel, then a x b = 0
• a x ka = 0 ??
– Null vector  [ 0 0 0 ]
– a  b | a || b | n̂ sin 
• n  unit vector perpendicular to the plane of a and b
• theta  the angle between them
–
–
–
–
–
a x b = - b x a (not commutative)
a x (b + c) = a x b + a x c
(ka) x b = a x (kb) = k (a x b)
i x j = k, j x k = i, k x i =j
axa=0
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Elements of Vector Geometry
• Lines
– a line through some point p0 and parallel to another vector t
 x
 x0 
t x 
– vector equation p(u )  p 0  ut
 y    y   u t 
 
 0
 y


• u  a scalar variable multiplying t 
z

 z0 

t z 

– Ordinary algebraic form
ut
x  x0  ut x
y
P
y  y0  ut y
z  z 0  ut z
x
– a line through two given points p0 and p1
p  p 0  u (p1  p 0 )
0 <= u <= 1 ??
P0
z
x  x0  u ( x1  x0 )
y  y0  u ( y1  y0 )
z  z0  u ( z1  z0 )
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Elements of Vector Geometry
•
Planes
– four ways to define a plane using vector equation
1. a plane through p0 and parallel to two independent vectors s and t
p  p0  us  wt
P0
wt
2. Three points p0, p1, and p2 (not collinear)
P
p  p 0  u (p1  p 0 )  w(p 2  p1 )
•
Normal vector  any vector perpendicular to a plane
n  st
•
us
n  (p1  p 0 )  (p 2  p1 )
Unit normal vector
n
nˆ 
|n|
P0
P1
u(p1-p0)
w(p2-p1)
P2
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Elements of Vector Geometry
3. A plane is by using a point it pass through and the normal vector
to the plane
(p  p 0 )  nˆ  0
– The scalar product of two mutually perpendicular vectors is zero
Ax  By  Cz  d  0
nˆ x x  nˆ y y  nˆ z z  d  0
P0
4. variation of the third way
– Given vector d
• a point on the plane
• perpendicular to the plane
(p  d)  d  0
n
d
n
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Elements of Vector Geometry
• Point a intersection b/w a plane and a straight line
– The plane pP
pP  a  ub  wc
– The straight line pL p L  d 0  ue
– Intersection point pP = pL
a  ub  wc  d  te
• solution for t
– Scalar product of both sides the equation with (b x c)
t
(b  c)  a  (b  c)  d
(b  c)  e
• solution for u ??  (c x e)
• solution for w ??  (b x e)
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