Lecture 2b -- Math with Vectors

1 11/21/20181 vector calculus :Math with VectorsEE3321 Electromagnetic Field TheoryCourse InstructorDr. Raymond C. RumpfOffice: A-337 Phone: (915) 747-6958E-Mail: vector Addition and Subtraction Dot Product Projections Cross Product Area vector Algebra Rules Triple ProductsVector calculus -- Math with VectorsSlide 211/21/20182 Visualization of vector Addition & SubtractionVector calculus -- Math with VectorsVector AdditionVector SubtractionU V UV xyU V xyUV Slide 3 vector Addition & SubtractionVector calculus -- Math with VectorsCartesianCylindricalSpherical xxyyzzxxyyzzUUaUaUaVVaVaVa Starting Vectors xxxyyyzzzUVUVaUVaUVa Addition xxxyyyzzzUVUVaUVaUVa Subtraction zzzzUUaUaUaVVaVaVa Starting Vectors zzzUVUVaUVaUVa Addition zzzUVUVaUVaUVa Subtraction rrrrUUaUaUaVVaVaVa Starting Vectors rrrUVUVaUVaUVa Addition rrrUVUVaUVaUVa SubtractionSlide 411/21/20183

2 The Dot Product, vector calculus -- Math with VectorsAB The dot product is all about projections. That is, calculating how much of one vector lies in the direction of another A B Slide 5 Projection of ontoVector calculus -- Math with VectorsA B 2 DirectionMagnitude BBBABBBBABAABBBB A B BA Slide 611/21/20184 Projection of ontoVector calculus -- Math with VectorsB A A B AB 2 DirectionMagnitude AAABAAAABABBAAAA Slide 7 The Dot Product TestVector calculus -- Math with VectorsThe dot produce can be used to test of two Vectors are perpendicular. If they are, the component of one along the other must be zero so the dot product must be when ABAB Slide 811/21/20185 The Cross Product, vector calculus -- Math with VectorsAB The cross product is all about area and calculating Vectors that are perpendicular to and.

3 SinnABABa A B B A 90 90 AreaAB is a unit vector perpendicularto the plane defined by and .naAB AB Slide 9 Handedness of the Cross ProductVector calculus -- Math with VectorsSlide 1011/21/20186 Calculating Cross Products (1 of 2) vector calculus -- Math with VectorsSuppose we wish to calculate the cross product .AB xyzxyxyzxyxyzxyAAAAABBBBB xyzAAxAyAz xyzBBxByBz Step 1 Construct an augmented two columns are repeated outside of the 11 Calculating Cross Products (2 of 2) vector calculus -- Math with VectorsStep 2 Multiply elements along the diagonals. yxzyxzABzABxABy xyzxyzABzAByABxStep 3 Make left-hand side products negative. Step 4 Add up all of the products. yzzyzxxzxyyxABABABxABAByABABz Slide 1211/21/20187 The Cross Product TestVector calculus -- Math with VectorsThe cross product can be used to test if two Vectors are parallel.

4 If they are, the cross product will be zero because the angle between the Vectors is when ||ABAB Slide 13 vector Algebra RulesVector calculus -- Math with VectorsCommutative LawsABBA Distributive Laws ABCABAC Self-Product2 AAA ABBA ABCABAC 0AA Associative Laws ABCABC Slide 1411/21/20188 vector Triple ProductsVector calculus -- Math with VectorsScalar Triple ProductThe scalar triple product is the volume of a parallelpiped. BCACABABC vector Triple ProductThe vector triple product arises when deriving the wave equation. ABCBACCAB C B A AB Slide 15