1. VECTOR

1. 1. VECTOR 2006. 9. 류승택

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. 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. 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) • Direction • Ex) result = tan theta theta = arctan result

5. 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

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. 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) • Unit vector • Any vector whose length is equal to one • Ex) Direction cosine of a

8. Scalar Multiplication • Multiplying any vector a by a scalark 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. 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)

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

11. a b 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.

12. V (unit vector) W X Scalar Product • 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

13. Scalar Product (참조) • 풀이과정

14. a b 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 sc = 0 ?? b sc = 0 ??

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. Elements of Vector Geometry • Lines • a line through some point p0 and parallel to another vector t • vector equation • u a scalar variable multiplying t • Ordinary algebraic form • a line through two given points p0 and p1 ut P0 y P x z 0 <= u <= 1 ??

17. Elements of Vector Geometry • Planes • four ways to define a plane using vector equation • a plane through p0 and parallel to two independent vectors s and t • Three points p0, p1, and p2 (not collinear) • Normal vector any vector perpendicular to a plane • Unit normal vector us P0 wt P u(p1-p0) P0 P1 w(p2-p1) P2

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 P0 n d n

19. Elements of Vector Geometry • Point a intersection b/w a plane and a straight line • The plane pP • The straight line pL • Intersection point pP = pL • 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)