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CS 335 Graphics and Multimedia Matrix Algebra Tutorial Properties of Vector Cross Product Cross product of parallel vectors r V1 ×V2 = V2 ×V1 = 0 iff V1 parallelto V2 Anti-commutative V1 ×V2 = −(V2 ×V1 ) Not associative V1 × (V2 ×V3 ) ≠ (V1 ×V2 ) ×V3 Distributive with respect to vector addition V1 × (V2 + V3 ) = (V1 ×V2 ) + (V1 ×V3 ) Basic Definitions Scalar – a number e.g. 3.1, 2.9, 10.7e5 Vector – a column (or a row) of scalars ⎡a ⎤ 3 ⎡ ⎤ ⎢b ⎥ V = [u , v, s, t ], V = ⎢⎢4⎥⎥, V = ⎢ ⎥ ⎢c ⎥ ⎢⎣7 ⎥⎦ ⎢ ⎥ ⎣d ⎦ Matrix – an array of numbers (or a collection of vectors) ⎡3.2 9.4⎤ M =⎢ , ⎥ ⎣ 5 7.1⎦ ⎡a N = ⎢⎢ e ⎢⎣ x b c f g y z d⎤ i ⎥⎥ w⎥⎦ Vector Math Add ⎡ x1 ⎤ ⎡ x2 ⎤ V1 = ⎢⎢ y1 ⎥⎥,V2 = ⎢⎢ y2 ⎥⎥ ⎢⎣ z1 ⎥⎦ ⎢⎣ z2 ⎥⎦ ⎡ x1 + x2 ⎤ V1 + V2 = ⎢⎢ y1 + y2 ⎥⎥ ⎢⎣ z1 + z2 ⎥⎦ Scale ⎡ax⎤ aV = ⎢⎢ay⎥⎥ ⎢⎣az⎥⎦ Transpose T ⎡ x⎤ ⎢ ⎥ T V = ⎢ y⎥ = [x ⎢⎣ z ⎥⎦ y z] Vector Math Dot product V1 ⋅V2 = V1 V2 cosθ where θ is the angle between the two vectors V2 θ |V2| cos θ V1 ⋅ V2 = V1 xV2 x + V1 yV2 y + V1 zV2 z V1 ⋅ V2 = V2 ⋅ V1 = 0 if and only if V1 ⊥ V2 V1 ⋅ (V2 + V3 ) = V1 ⋅ V2 + V1 ⋅ V3 V1 Vector Product (cross product) Multiplication of two vectors to produce another vector. V1 ×V2 = u V1 V2 sinθ Where u is a unit vector that is perpendicular to both V1 and V2 V 1× V 2 (Vector Notation) u V2 V1 ⎛V1yV2 z −V1zV2 y ⎜ V1 ×V2 = ⎜V1zV2 x −V1xV2 z ⎜ ⎜V V −V V ⎝ 1x 2 y 1 y 2 x ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ cross product gives a vector in a V1 ×V2 = (V1yV2 z −V1zV2 y ,V1zV2 x −V1xV2 z ,V1xV2 y −V1yV2 x ) direction perpendicular to the two original vectors (u) and with a magnitude equal to the area of Nice applet demo at: the shaded parallelogram http://physics.syr.edu/courses/java-suite/crosspro.html Matrix Algebra ⎡x y ⎤ ⎡a b ⎤ M =⎢ ,N = ⎢ ⎥ ⎥ ⎣ z w⎦ ⎣c d ⎦ Matrix + Matrix (add) ⎡a + x b + y ⎤ M +N =⎢ ⎥ + + c z d w ⎣ ⎦ Transpose ⎡a c ⎤ M =⎢ ⎥ ⎣b d ⎦ T Watch the dimensions!! Scale ⎡ sa sb ⎤ sM = ⎢ ⎥ ⎣ sc sd ⎦ Matrix * Matrix (multiplication) ⎡ax + bz ay + bw⎤ MN = ⎢ ⎥ ⎣ cx + dz cy + dw⎦ Matrix Inverse M’s inverse, denoted as M-1, is a matrix such that ⎡1 0 0 0 ⎤ ⎢0 1 0 0 ⎥ ⎥ MM −1 = M −1M = I = ⎢ ⎢ ... ⎥ ⎢ ⎥ ⎣0 0 0 1 ⎦ Matrix Algebra Matrix * Vector ⎡a b ⎤ M =⎢ , ⎥ ⎣c d ⎦ ⎡ax + by ⎤ MV = ⎢ ⎥ ⎣ cx + dy ⎦ ⎡ x⎤ V =⎢ ⎥ ⎣ y⎦ Take-home Exercise ⎡a b ⎤ ⎡1 2 0 ⎤ ⎡3 0 2⎤ ⎡ x⎤ A = ⎢⎢0 c ⎥⎥, M = ⎢⎢0 4 8 ⎥⎥, N = ⎢⎢1 2 4⎥⎥, V = ⎢⎢ y ⎥⎥ ⎢⎣1 0⎥⎦ ⎢⎣4 1 2⎥⎦ ⎢⎣0 0 1 ⎥⎦ ⎢⎣ 1 ⎥⎦ compute : MN = ( AM ) N = ( AT M ) N = AT ( MN ) = NV =, VTN = VV T = V TV =