Lecture_Chapter4_a

Space can be uniquely represented by a linear

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Unformatted text preview: = 2 ￿￿ 3 v2 = 0 l cos α = ￿v1 ￿ v2 l = ￿v1 ￿ cos α = v1 · ￿v 2 ￿ ECS 175 Chapter 4: Object Transformation 15 Orthogonal Vectors •  Orthogonal vectors •  Enclose a right angle (dot product is zero) •  Are linearly independent •  May be used to create an orthogonal basis/coordinate frame •  Vectors is this space can be uniquely represented by a linear combination v= n ￿ v i bi bi base vectors i ECS 175 Chapter 4: Object Transformation 16 Orthogonal Vectors •  Orthogonal vectors in 2D ￿￿ x v= y v⊥1 = ￿ y −x ￿ v⊥2 = ￿ −y x ￿ •  Orthogonal vectors in 3D x v = y z w = v ⊥1 y = −x if x,y not both equal to zero 0 Create third orthogonal vector with the help of cross product ECS 175 Chapter 4: Object Transformation 17 Orthogonal Vectors and Area •  The cross product v2 w3 − v3 w 2 u = v × w = v3 w1 − v1 w3 v1 w2 − v2 w 1 ￿v × w￿ = A u·v =0 u·w =0 0 For 2D vectors v,w: u = 0 A ECS 175 Chapter 4: Object Transformation 18 How...
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This document was uploaded on 03/12/2014 for the course ECS 175 at UC Davis.

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