General (2A
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Vector Rotation (1 Rot (θ (x 1, y 1 (x 0, θ x 0 General (2A 3
Vector Rotation (2 (x 1, y 1 (x 0, y 1 (x 1, y 1 (x 0, R R x 1 x 0 rotate by θ (x 0, (x 1, y 1 x 1 y 1 = x 0 cos θ sin θ = x 0 sin θ + cosθ General (2A 4
Vector Rotation (3 (x 1, y 1 (x 0, x 0 x 0 In the rotated coordinate invariant length x 0, General (2A 5
Vector Rotation (4 x 1 = x 0 cosθ sin θ y 1 = x 0 sin θ + cosθ (x 1, y 1 (x 1, y 1 (x 0, x 0 sin θ x 0 (x 0, y 1 x 0 cos θ x 0 sin θ sin θ x 1 sin θ General (2A 6
Transformation v = (x 0, Rot (θ = (x 1, y 1 rotation operator V W v f : V W W transformation map General (2A 7
Matrix Transformation v (x 0, f : V W (x 1, y 1 transformation x 1 y 1 = x 0 cos θ sin θ = x 0 sin θ + cosθ [ x 1 1] y = [ cosθ sin θ ][ x 0 1] sin θ cosθ y [ A = cosθ sin θ ] sin θ cosθ = A = T A ( x x x T A General (2A 8
Coordinates and Coordinates Systems Rectangular Coordinate System (1.5, 1.5 any vector (x, y coordinates (1.5, 1.5 Non-Rectangular Coordinate System any vector 3 2 2 ( 3 2 2, 3 2 (x ', y ' coordinates ( 3 2 2, 3 2 General (2A 9
Standard Basis 1 e 2 R 2 e 1 e 1 = (1, 0 e 2 = (0, 1 R 2 any vector (x, y 1 standard basis {e 1, e 2 } spans R 2 1 1 e 3 R 3 e 2 e 1 = (1, 0, 0 e 2 = (0, 1, 0 e 3 = (0, 0, 1 (x, y R 3 any vector e 1 1 standard basis {e 1, e 2, e 3 } spans R 3 General (2A 10
Basis and Coordinates (1.5, 1.0 = 1.5 e 1 + 1.0 e 2 V m 1 basis e 2 = 1.5[ 1 0] + 1.0 [ 0 1] m 2 e 1 basis {e 1, e 2 } coordinates (1.5, 1.0 = [1.5 1.0] [ 1 0 0 1] collinear vectors linearly dependent vectors many bases but the same number of basis vectors basis {u 1, u 2 } basis {, } basis { 1, 2 } coordinates coordinates coordinates u 2 (x u, y u (x v, y v 2 (x, y 1 u 1 General (2A 11
Standard Basis & Standard Matrix e 2 = (0, 1 f : V W T A (e 2 = ( sinθ, cos θ T A (e 1 = (cos θ, sinθ e 1 = (1, 0 transformation standard basis standard matrix [ x 1 1] y = [ cosθ sin θ ][ x 0 1] sin θ cosθ y T A ( e 1 T A ( e 2 T A ( e n = = A T A ( x x A = x T A General (2A 12
Dimension In vector space R 2 any one vector line R 1 linearly independent any to non-collinear vectors plane R 2 linearly independent any three or more vectors linearly dependent v 3 = k 1 + k 2 R 2 R 2 R 1 v 3 General (2A 13
Basis S = {,,, v n } non-empty finite set of vectors in V S S is a basis linearly independent k S spans V span(s = span{,,, v n } = k 1 + k 2 R 2 2 R all possible linear combination of the vectors in in S { = k 1 + k 2 + + k n v n } General (2A 14
Linear Dependent (1 {u, v, } linearly dependent = u + v u = v v = u v v v u u u u + v = 0 u + v = 0 u + v = 0 General (2A 15
Linear Dependent (2 {u, v, } linearly dependent v v v u u u k 1 u + k 2 v + k 3 = 0 m 1 u + m 2 v + m 3 = 0 n 1 u + n 2 v + n 3 = 0 (k 1 = 0 (k 2 = 0 (k 3 = 0 (k 1 0 (k 2 0 (k 3 0 (m 1 = 0 (m 2 = 0 (m 3 = 0 (m 1 0 (m 2 0 (m 3 0 (n 1 = 0 (n 2 = 0 (n 3 = 0 (n 1 0 (n 2 0 (n 3 0 General (2A 16
Linear Independent (1 S = {,,, v n } non-empty set of vectors in V k 1 + k 2 + + k n v n = 0 the solution of the above equation trivial solution: k 1 = k 2 = = k n = 0 if other solution exists if no other solution exists S linearly dependent S linearly independent v 3 k 2 k 3 v 3 v 3 v 3 k 1 General (2A 17
Linear Independent (2 S = {,,, v n } non-empty set of vectors in V k 1 + k 2 + + k n v n = 0 the solution of the above equation k 1 = k 2 = = k n = 0 if other solution exists if no other solution exists S S linearly dependent linearly independent at least one vector in S is a linear combination of the other vectors in S v i = k 1 + k 2 + + k i 1 v i 1 + k i+1 v i+ 1 + + k n v n S linearly dependent no vector in S is a linear combination of the other vectors in S v i = k 1 + k 2 + + k i 1 v i 1 + k i+1 v i+ 1 + + k n v n S linearly independent General (2A 18
Linear Independent (3 S = {,,, v n } non-empty set of vectors in V k 1 + k 2 + + k n v n = 0 the solution of the above equation k 1 = k 2 = = k n = 0 if other solution exists S linearly dependent if no other solution exists S linearly independent S = { 0 } linearly dependent S = { } linearly independent S = {, } k linearly independent General (2A 19
Change of Basis u 2 = (0, 1 u' 2 v = (k 1, k 2 u 1 = (1, 0 u' 1 Old Basis B = {u 1, u 2 } B ' = {u' 1, u' 2 } Ne Basis [u ' 1 ] B = [ cosθ sin θ ] coordinate of u' 1 ith respect to B u' 1 = cosθ u 1 + sin θu 2 u' 2 = sin θu 1 + cosθu 2 [u ' 2 ] B = [ sin θ cosθ ] coordinate of u' 2 ith respect to B v = k 1 u' 1 + k 2 u' 2 = k 1 (cosθu 1 +sin θ u 2 + k 2 ( sin θ u 1 +cosθu 2 = (k 1 cosθ k 2 sin θu 1 +(k 1 sin θ+k 2 cosθu 2 [v ] B' = [ k 1 k 2] coordinate of v ith respect to B' [v ] B = [ k 1cos θ k 2 sinθ k 1 sinθ+k 2 cos θ ] coordinate of v ith respect to B General (2A 20
Change of Basis u 2 = (0, 1 u' 2 v = (k 1, k 2 u 1 = (1, 0 u' 1 Old Basis B = {u 1, u 2 } B ' = {u' 1, u' 2 } Ne Basis [v ] B' = [ k 1 k 2] coordinate of v [ ith respect to B' [v ] B = cosθ sin θ][ k 1 2] sinθ cos θ k coordinate of v ith respect to B [u ' 1 ] B = [ cosθ sin θ ] coordinate of u' 1 ith respect to B [v ] B = [ cos θ sinθ] sin θ cosθ [ v] B ' [v ] B = P B ' B [v ] B ' [u ' 2 ] B = [ sin θ cosθ ] coordinate of u' 2 ith respect to B P B ' B = [ [u' 1 ] B [u' 2 ] B ] General (2A 21
Transition Matrix P B ' B = [ [u' 1 ] B [u' 2 ] B [u' n ] B ] [v ] B = P B ' B [v ] B ' [u ' 1 ] B [u ' 2 ] B coordinate of u' 1 ith respect to B coordinate of u' 2 ith respect to B [v ] B' coordinate of v ith respect to B' [v ] B coordinate of v ith respect to B P B B' = [ [u 1 ] B' [u 2 ] B' [u n ] B ' ] [v ] B' = P B B ' [v ] B [u 1 ] B ' [u 2 ] B' coordinate of u 1 ith respect to B' coordinate of u 2 ith respect to B' [v ] B coordinate of v ith respect to B [v ] B' coordinate of v ith respect to B' General (2A 22
Change of Basis e 2 = (0, 1 f : V W T A (e 2 = ( sinθ, cos θ T A (e 1 = (cos θ, sinθ e 1 = (1, 0 transformation Old Basis Ne Basis [ x 1 1] y = [ cosθ sin θ ][ x 0 1] sin θ cosθ y T A ( e 1 T A ( e 2 T A ( e n = A = T A ( x x A = x T A General (2A 23
Transformation v (x 0, f : V W (x 1, y 1 transformation e 2 = (0, 1 f : V W T A (e 2 = ( sin θ, cos θ e 1 = (1, 0 transition T A (e 1 = (cosθ, sin θ Old Basis Ne Basis General (2A 24
Transformation (x 1, y 1 (x 0, (x 0, (x 0, e 2 = (0, 1 f : V W e 1 = (1, 0 transition Old Basis Ne Basis General (2A 25
Vector Rotation (2 (x 1, y 1 (x 0, x 0 In the rotated coordinate invariant length x 0, General (2A 26
Trasformation x 1 = x 0 cosθ sin θ y 1 = x 0 sin θ + cosθ (x 1, y 1 (x 1, y 1 (x 0, x 0 sin θ x 0 (x 0, y 1 x 0 cos θ x 0 sin θ sin θ x 1 sin θ General (2A 27
Normal Vector & 3 Points z z a b = i j k a 1 a 2 a 3 3 b 1 b 2 b i = a 2 a 3 3 j a 1 a 3 3 k + a 1 a 2 2 b 2 b b 1 b b b 1 General (2A 28
Normal Vector & 3 Points z Non-collinear 3 points z General (2A 29
References [1] http://en.ikipedia.org/ [2] http://planetmath.org/ [3] M.L. Boas, Mathematical Methods in the Physical Sciences [4] D.G. Zill, Advanced Engineering Mathematics