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# Vectors Vectors Vectors * Scalars v=Q-P Vector from P to Q

Example Vectors Geometric Vectors Directed Line Segments Scalar-Vector Multiplication Changes Length Maintains Direction Vector-Vector Addition Head-to-Tail Axiom

Euclidean Spaces Inner (dot) product u·v=v·u (au+bv)·w=au·w+bv·w v·v>0(v!=0) 0·0=0 if u·v=0, then u and v orthogonal |v|=sqrt(v·v) Add concepts of affine spaces, such as points P-Q is a vector |P-Q|=sqrt((P-Q)·(P-Q)) Angle between two vectors u·v=|u||v|cos(theta) Equation for a circle |Q-P|=r sqrt((Q-P)·(Q-P))=r (Q-P)·(Q-P)=r2

Dot Product v·w = w·v v·v==0 only when v==0

||v|| = sqrt(v·v) = Distance to v from origin

||v-w|| = Distance from v to w

Projections Finding the shortest distance between a
point to a line or plane Start with two vectors Divide one into two parts One parallel, one orthogonal to the second
original vector v is first vector, w is second vector  w=av+u Parallel part, v Orthogonal part, u u·v=0 w·v=av·v+u·v=av·v a=-(w·v)/(v·v) av is projection of w onto v u=w-[(w·v)/(v·v)]v

Normalized Vectors Length of One Unit vector v'=v/||v|| Angle Between v and w
Calulating the length of a rotated vector. The dot product
of v and w is the length of the projection of w onto v,
provided v is a unit vector
||u|| =||w||cos(theta)
=||w||((v·w)/||v||||w||) OpenGL and Affine Spaces Positions always defined relative to the
origin of the current coordinate system Transform the coordinate system to take
positions relative to some other point Necessary for computing distances, angles,
etc.

Matrices nxm array of scalars n Rows m Columns If m=n then square matrix The elements of a matrix A are {aij}, i=1,...,n,
j=1,...,n A=[aij] Transpose of A of n is mxn matrix with rows
and columns interchanged AT=[aji] Single row or single column Row Matrix Column Matrix b=[bi] Transpose of a row matrix is a column
matrix Transpose of a column matrix is a row
matrix Identity Matrix MI=M Identity Matrix 1's along diagonal I=[aij] aij={1 if i=j, 0 otherwise AI=A IB=B

Matrix Operations Scalar-matrix multiplication aA=[aaij] a(bA)=(ab)A abA=baA Matrix-matrix addition C=A+B=[aij+bij] Commutative Associative Matrix-matrix multiplication Product of nxl and lxm is nxm Middle dimmension MUST match C=AB=[cij] cij=Sum[k=1,l]aikbkj Associative NOT Commutative AB!=BA May in fact not even be defined Order matters Application to Transformations

Row and Column Matrices Representation of vectors or points p=[x y z]T pT=[x y z] Transformation - Square matrix Point - Column matrix of 2, 3, or 4 pts. p'=Ap Transforms p by A p'=ABCp Concatenations of transformations (AB)T=BTAT