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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.
 Spring '08
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