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Solution Vectors and points Can represent most abstract objects as combinations of these E.g. lines, planes, polygonsL& m& m    ?What Is a Vector?    rGeometric object with two properties direction length (if length is 1, is unit vector) Graphically represented by H%2%% r    @Scale    VGraphically change length of vector v by aJ  +    AAddition    kGraphically put tail of second vector at head of first draw new vector from tail of first to head of second& ` `k    BAlgebraic Vectors    Algebraically, vectors are more than this Any entity that meets certain rules (lies in vector space) can be called  vector Ex: Matrices, quaternions, fixed length polynomials Mostly mean geometric vectors, however$W t4     A    ;x Vector Space&Set of vectors related by +, Meet rules v + w = w + v (commutative +) (v + w) + u = v + (w + u) (associative +) v + 0 = v (identity +) v + (-v) = 0 (inverse +) (ab) v = a (bv) (associative ) (a+b)v = av + bv (distributive ) a(v + w) = av + aw (distributive ))ZZ "  ## #4 Number SpacesCardinal  Positive numbers, no fractions Integer  Pos., neg., zero, no fractions Rational  Fractions (ratios of integers) Irrational  Non-repeating decimals (p,e) Real  Rationals+irrationals Complex  Real + multiple of -1 a+bi : 7aReal Vector SpacesLUsually only work in these Rn is an n-dimensional system of real numbers Represented as ordered list of real numbers (a1,& ,an) R3 is the 3D world, R2 is the 2D world J6' %,    >  +  8&  <yLinear CombinationCombine set of n vectors using addition and scalar multiplication v = a1v1 + a2v2 + & + anvn Collection of all possible linear combinations for given v1& vn is called a span Linear combination of 2 perpendicular vectors span a planerBZZZ29 ;6Y=M=zLinear Dependence A system of vectors v1, & ,vn is called linearly dependant if for at least one vi vi = a1v1 +& + ai-1vi-1 + ai+1vi+1 +& + anvn Otherwise, linearly independent Two linearly dependant vectors are said to be collinear I.e. w = a.v I.e. they point the  same directionRZ+ZXZ2Z    . %,\ELinear Dependence?Example Center vector can be constructed from outer vectors`B Vector Basis     pOrdered set of n lin. ind. vectors b = { v1, v2, & , vn} Span n-dimensional space Represent any vector as linear combo v = a1v1 + a2v2 + & + anvn Or just components v = (a1, a2, & , an)$>8                              >          >{Vector Representation    3D vector v represented by (x, y, z) Use standard basis { i, j, k } Unit length, perpendicular (orthonormal) v = xi + yj + zk Number of units in each axis direction%ZZZ'Z )'a                 '    fDVector Operations    2Addition: +,- Scale: Length: ||v|| Normalize: D3  2    C}Addition Add a to b N "     D|Scalar Multiplication<change length of vector v by a<  E~LengthLength ||v|| gives length (or Euclidean norm) of v if ||v|| is 1, v is called unit vector usually compare length squared Normalize v scaled by 1/||v|| gives unit vector s '     HVector OperationsGames tend to use most of the common vector operations Addition, Subtraction Scalar multiplication Two others are extremely common: Dot product Cross product`7,!7,  =  Dot product     )Also called inner product, scalar product)    ?"Dot Product: Uses    a " a equals ||a||2 can test for collinear vectors if a and b collinear & unit length, |a " b| ~ 1 Problems w/floating point, though can test angle/visibility a " b > 0 if angle < 90 a " b = 0 if angle = 90 (orthogonal) a " b < 0 if angle > 903ZSZZYZ )!    FDot Product: Example    Suppose have view vector v and vector t to object in scene (t = o - e) If v " t < 0, object behind us, don t drawbr  #q    >!Dot Product: Uses    Projection of a onto b isD     GDot Product: Uses    AExample: break a into components collinear and perpendicular to b:B/A    A$ Cross Product     qCross product: definition returns vector perpendicular to a and b right hand rule length = area of parallelogramRX!0q    B%Cross Product: Uses    pgives a vector perpendicular to the other two! ||a b|| = ||a|| ||b|| sin(q) can test collinearity ||a b|| = 0 if a and b are collinear Better than dot  don t have to be normalizeddU1 =4W     U    IOther Operations    fSeveral other vector operations used in games may be new to you: Scalar Triple Product Vector Triple Product These are often used directly or indirectly in game code, as we ll see<A,GA,G    PScalar Triple Product_Dot product/cross product combo Volume of parallelpiped Test rotation direction Check sign DT T B+      #    C&Triple Scalar Product: Example    aCurrent velocity v, desired direction d on xy plane Take If > 0, turn left, if < 0, turn rightvb4+    4    ZVector Triple ProductRTwo cross products Useful for building orthonormal basis Compute and normalize: P:: 4(         8Points    Points are positions in space  anchored to origin of coordinate system Vectors just direction and length  free-floating in space Can t do all vector operations on points But generally use one class in library    /Point-Vector Relations    5Two points related by a vector (Q - P) = v P + v = Q 5    [; Affine Space(      xVector, point related by origin (P - O) = v O + v = P Vector space, origin, relation between them make an affine space C64l        aCCartesian Frame    Basis vectors {i, j, k}, origin (0,0,0) 3D point P represented by (px, py, pz) Number of units in each axis direction relative to origin    ;hC            ;    ^Affine CombinationBLike linear combination, but with points P = a1P1 + a2P2 + & + anPn a1,& ,an barycentric coord., add to 1 Same as point + linear combination P = P1 + a2 (P2-P1) + & + an (Pn-P1) If vectors (P2-P1), & , (Pn-P1) are linearly independent, {P1, & , Pn} called a simplex (think of as affine basis) )Z@Z#Z%ZqZ)  #  t? M' _Convex CombinationAffine combination with a1,& ,an between 0 and 1 Spans smallest convex shape surrounding points  convex hull Example: triangle j  b1Points, Vectors in Games    Points used for models, position vertices of a triangle Vectors used for velocity, acceleration indicate difference between points, vectorsL!(,!(,    0Parameterized Lines    Can represent line with point and vector P + tv Can also represent an interpolation from P to Q P + t(Q-P) Also written as (1-t)P + tQ*)0') P-      T      D'Planes    G2 non-collinear vectors span a plane Cross product is normal n to planeJH G    \PlaneslDefined by normal n = (A, B, C) point on plane P0 Plane equation Ax+By+Cz+D = 0 D=-(AP0x + BP0y + CP0z) V '+ ! ! ! .   (#]Planes_Can use plane equation to test locality of point If n is normalized, gives distance to plane&`7( References     Anton, Howard and Chris Rorres, Elementary Linear Algebra, 7th Ed, Wiley & Sons, 1994. 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