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Advanced Graphics Intro_Part_3 - Basis I I I in R3 a basis...

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Basis in R 3 a basis is a set of 3 non-parallel vectors common to use orthonormal basis basis vectors are mutually orthogonal basis vectors have unit magnitude basis most students are familiar with [ ] [ ] [ ] 1 0 1 0 0 0 1 0 0 0 1 = = = = = = = = = k j i k j k i j i k j i T T T H H H H H H H H H H H H Direction (cont) can write any vector as a linear combination of basis vectors k j i H H H 7 2 3 7 2 3 + + = ú ú ú û ù ê ê ê ë é

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Cross Product Hill 4.4 only defined in R 3 defined in terms of standard basis [ ] [ ] c k b a b a j b a b a i b a b a b b b a a a b a x y y x z x x z y z z y T z y x T z y x H H H H H H = + + = × = × ) ( ) ( ) ( Cross Product (cont) cross product of two vectors is a vector that is orthogonal to the original two vectors direction given by right hand rule
Cross Product (cont) properties j i k i k j k j i b a s b a s c a b a c b a a b b a H H H H H H H H H H H H H H H H H H H H H H H H = × = × = × × = × × + × = + × × = × y homogeneit ) ( ) ( linearity ) ( ry antisymmet

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Advanced Graphics Intro_Part_3 - Basis I I I in R3 a basis...

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