SJSU EE 140 - ch2_Vector_Algebra (16 pages)

Chapter 2 Vector Algebra Review Dr Ray Kwok SJSU Vector Algebra Dr Ray Kwok Vector products scalar scalar scalar scalar a b ab e g 2 4 kg 8 kg scalar vector vector e g 5 2 x 4 y 10 x 20 y r r k A kA vector times vector r r can be scalar scalar product or dot product A B or r r vector vector product or cross product A B Can you add a vector to a scalar Vector Algebra Dr Ray Kwok The scalar product dot product r r A B AB cos x x 1 y y z z x y 0 x z y z r r A B A x x A y y A z z Bx x B y y Bz z r r A B A x B x A y B y A z Bz e g r A 2x 2 y r B 3x y What is A B e g What s the angle between these 2 arrows Vector Algebra Dr Ray Kwok Interpretation projection Vector Algebra Dr Ray Kwok Example Work r r W F x Projection So is the uplifting force doing anything at all Vector Algebra Dr Ray Kwok The vector product cross product r r A B AB sin x x 0 y y z z x y z right hand coordinate y z x cyclic permutation z x y y x z z y x x z y e g r A 2x 2 y r B 3x y What is A x B x r r A B Ax y z Ay Az Bx By Bz e g What s the angle between these 2 arrows Vector Algebra Dr Ray Kwok Example calculate torque r r r F r 4N x o 40o 1 5 m choose counter clockwise o 1 5 4 sin 40o 3 86 N m counter clockwise Vector Algebra Dr Ray Kwok Example perpendicular F 4N 40o x 1 5 m o 4 sin 40o 1 5 m x o o 1 5 4 sin 40o 3 86 N m counter clockwise Vector Algebra Dr Ray Kwok Example moment arm 4N 40o x 1m o 1 5 m 1 5 sin 40o o 40 x o 4N 1 5 m o 4 1 5 sin 40o 3 86 N m counter clockwise Vector Algebra Dr Ray Kwok Exercise 1 Find a b c d e f r r A B r r B 2A r r A B r r A B r r A A r r B B r A x 4 z r B 2 x y z g Angle between A and B h Find a vector that is perpendicular to A and B Vector Algebra Dr Ray Kwok Triple vector product scalar vector Ax r r r A B C Bx Cx Ay Az By Cy Bz Cz r r r r r r r r r r r r A B C A C B B C A A B C homework Vector Algebra Dr Ray Kwok OCC Orthogonal Orthogonal Curvilinear Coordinates e i e j ij Curvilinear coordinate surfaces can be curved Cartesian Cylindrical Spherical coordinates Transformation between coordinates Line Area Volume integral in each coordinate Vector Algebra Dr Ray Kwok Rectangular Polar 2D Polar to rectangular x r cos y r sin Rectangular to polar r2 x2 y2 tan y x Pythagorean theorem be certain which angle is Vector Algebra Dr Ray Kwok Transformation Vector Algebra Dr Ray Kwok Vector operations Vector Algebra Dr Ray Kwok Homework Ch 2 3 5 10 12 13 15 20 23 26 30 and prove eqn 2 33 Also prove that r r r r r r r r r A B C A C B B C A r r r r r r r r r r r r A B C D A C B D A D B C