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# vectors - Physics 2306 M Blecher Chapter 1 Vectors 1.1...

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Physics 2306 M. Blecher August 18, 2009

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Chapter 1 Vectors 1.1 Representation of Vectors Vectors in three dimensional (3D) space are quantities possessing magnitude and direction. They are written in bold font in this note. In component form,vector A is written, A = 3 summationdisplay i =1 A i e i , (1.1) where A i is the component of the vector along the direction of the unit vector e i . The units [A] of the vector go with each component. Thus if A is a force, [ A i ]=Newtons (N). The unit vectors have magnitude of one, do not possess units, and specify one of the three mutually perpendicular directions of a right-handed system of axes in 3D. Thus unit vectors obey the scalar and vector product rules, e i · e j = δ ij , (1.2) e i × e j = 3 summationdisplay k =1 ǫ ijk e k . (1.3) In Equation 1.2, δ ij = (0 , 1) if ( i negationslash = j, i = j ). In Equation 1.3, ǫ ijk = (0 , ± 1) if (two or three indices are equal, indices all different). By definition ǫ 123 1 and an interchange of two indices causes multiplication by -1. With these 1
rules the scalar and vector products of any two vectors are, A · B = 3 summationdisplay i =1 A i B i = AB cos γ, (1.4) A × B = 3 summationdisplay ijk =1 ǫ ijk A i B j e k = AB sin γ n , (1.5) where A = radicalBig 3 i =1 A 2 i , B = radicalBig 3 i =1 B 2 i , γ is the angle between the vectors, and n A , B .

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vectors - Physics 2306 M Blecher Chapter 1 Vectors 1.1...

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