1. 1 1. VECTOR 2006. 9. 류승택

2. 2 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

3. 3 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

4. 4 Geometric Interpretation Geometric properties –Displacement First number : east if plus(+), or west if minus(-) Second number: north (+) or south (-) 3D 인경우 : Third number: up(+) or down(-) Ex) (16.3, -10.2) : 16.3(east), -10.2(south) –Distance Pythagorean theorem Ex) –Direction Ex) result = tan theta theta = arctan result

5. 5 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 = [ a 1 a 2 ] –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

6. 6 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

7. 7 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 = [a x a y a z ]  a = a x + a y + a z = a x i+ a y j + a z k Reverse the direction of any vector –multiplying each component by -1 Magnitude –Positive scalar –Ex) Unit vector –Any vector whose length is equal to one –Ex) Direction cosine of a

8. 8 Scalar Multiplication Multiplying any vector a by a scalar k  ka –Ex) Magnitude ka 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

9. 9 Vector Addition Given a = [a x a y a z ] and b = [b x b y b z ]] –a + b = [a x +b x a y +b y a z +b z ] 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)

10. 10 Scalar and Vector Products Multiply two vectors  Two different ways –Scalar product Produce a single real number –Vector product Produce a vector

11. 11 Scalar Product Scalar Product (= dot product) ( 내적 ) –The sum of the products of their corresponding components –Using the law of cosine, the angle between two vectors a and b satisfies the equation –Scalar Product Properties If a is perpendicular to b, then Scalar ?? A quantity that is completely specified by its magnitude and has no direction. a b

12. 12 Scalar Product –Use the dot product to project a vector onto another vector 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 W V (unit vector) X

13. 13 Scalar Product ( 참조 ) 풀이과정

14. 14 Vector Product Vector Product (= Cross Product) 외적 –c = a x b  c is perpendicular to both a and b Perpendicular to the pane defined by a and b a  c = 0 ?? b  c = 0 ?? a b

15. 15 Vector Product –If two vectors a and b parallel, then a x b = 0 a x ka = 0 ?? –Null vector  [ 0 0 0 ] – 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 –a x a = 0

16. 16 Elements of Vector Geometry Lines –a line through some point p 0 and parallel to another vector t –vector equation u  a scalar variable multiplying t –Ordinary algebraic form –a line through two given points p 0 and p 1 0 <= u <= 1 ?? x z y P0P0 P utut

17. 17 Elements of Vector Geometry Planes –four ways to define a plane using vector equation 1.a plane through p 0 and parallel to two independent vectors s and t 2.Three points p0, p1, and p2 (not collinear) Normal vector  any vector perpendicular to a plane Unit normal vector P0P0 P1P1 P2P2 u(p 1 -p 0 ) w(p 2 -p 1 ) P0P0 P usus wtwt

18. 18 Elements of Vector Geometry 3. A plane is by using a point it pass through and the normal vector to the plane –The scalar product of two mutually perpendicular vectors is zero 4. variation of the third way –Given vector d a point on the plane perpendicular to the plane P0P0 n n d

19. 19 Elements of Vector Geometry Point a intersection b/w a plane and a straight line –The plane p P –The straight line p L –Intersection point p P = p L solution for t –Scalar product of both sides the equation with (b x c) solution for u ??  (c x e) solution for w ??  (b x e)