1 1. scalar Physical concepts that require only one numerical quantity for their complete specification are scalar quantities. 2. vector Vector quantities require for their complete specification a positive quantity, called the magnitude of the vector and the direction. Notice: Not all things with a magnitude and direction are vectors. §2.1 scalar and vector quantities 2 §2.1 scalar and vector quantities Vector A r magnitude AA or r Direction 3. Representation of the vector §2.2 Operation of the vectors 1. Multiplication of a vector by a scalar AB rr α= 0>α 0<α A r B r A r B r Letter symbol arrow symbol 2. Vector addition by geometric methods A r B r Triangle rule or parallelogram rule: Polygon rule: 4321 RRRRR rrrrr +++= 1 R r 2 R r 3 R r 4 R r R r §2.2 operation of the vectors A r B r C r B r BAC rrr += Resultant: Note: BAC BABA +≠ +≠+ rr 3 3. Vector difference by geometric methods The operation +(- ) is identical in all respects to the operation implied by - 1 r r 1 r r A r B r )( BABAC rrrrr ?+=?= A r B r ? C r 4. The scalar(dot) product of two vectors The scalar product is a way to multiply two vectors to yield a scalar result. Define: θcosABBA =? rr §2.2 operation of the vectors Note: BACABBA ?≠?≠? rrrr A r B r θ §2.2 operation of the vectors 5. The cross product of two vectors The new vector that results from the vector product has both a magnitude and a direction. Define: BAC rrr ×= 1 A, B are always positive 2θ < π/2 3θ > π/2 4θ = π/2 5 0cos >=? θABBA rr 0cos <=? θABBA rr 0cos ==? θABBA rr CABACBA rsrrrrr ?+?=+? )( Notice: 4 BAC rrr ×= §2.2 operation of the vectors Magnitude: θsinABBA =× rr Direction:perpendicular to the plan containing the BA rr and Right hand rule for the direction of the cross product: A r B r A r B r Note: §2.2 operation of the vectors 1θ is always less than π in ABsin θ 2 3 4 5 0or0 =×=× BBAA rrrr ABBA rrrr ×≠× ABBA rrrr ×?=× CABACBA rrrrrrr ×+×=+× )( §2.3 The Cartesian representation of any vector 1. The Cartesian coordinate system Right handed Cartesian coordinate system: 5 §2.3 the Cartesian representation of any vector Unit(basis) vectors: kji ? , ? , ? Magnitudes: 1 Directions: indicate the directions that correspond coordinates increase. i ? k ? j ? x y z 1 ?? =? ii 0 ?? =? ji 0 ?? =?ki 0 ?? =?ij 1 ?? =? jj 0 ?? =?kj 0 ?? =?ik 0 ?? =? jk 1 ?? =?kk Scalar product of the Cartesian unit vectors with each other §2.3 the Cartesian representation of any vector 2. The case of two dimensions jAiA jAiAA yx ? sin ? cos ?? θθ += += r A r x A y A x y i ? j ? θ Vector A r Magnitude: 22 yx AAA += Direction: decided by angle x y A A 1 tan ? =θ 6 §2.3 the Cartesian representation of any vector 3. The case of three dimensions A r iA x ? kA z ? jA y ? x y z kAjAiAA zyx ??? ++= r z y x AAkA AAjA AAiA ==? ==? ==? γ β α cos ? cos ? cos ? r r r We can prove that 2122221 )()( zyx AAAAAA ++=?= rr §2.3 the Cartesian representation of any vector 4. The operations of vectors in Cartesian coordination system Addition: kBAjBAiBA kBjBiBkAjAiABA xzyyxx zyxzyx ? )( ? )( ? )( ) ??? () ??? ( +++++= +++++=+ rr 1coscoscos 222 =++ γβα x y z γ β α A r 7 §2.3 the Cartesian representation of any vector kBAjBAiBA kBjBiBkAjAiABA zzyyxx zyxzyx ? )( ? )( ? )( ) ??? () ??? ( ?+?+?= ++?++=? rr Difference: Multiplication: kAjAiA kAjAiAA zyx zyx ??? ) ??? ( ααα αα ++= ++= r Scalar product of two vectors: zzyyxx zyxzyx BABABA kBjBiBkAjAiABA ++= ++?++=? ) ??? () ??? ( rr §2.3 the Cartesian representation of any vector cross product of two vectors: kBABA jBABAiBABA kBjBiBkAjAiABA xyyx zxxzyzzy zyxzyx ? )( ? )( ? )( ) ??? () ??? ( ?+ ?+?= ++×++=× rr zyx zyx BBB AAA kji BA ??? =× rr or Mnemonic: kjikji ?????? + - 8 §2.3 the Cartesian representation of any vector where 0 ?? =×ii kij ??? ?=× jik ??? =× kji ??? =× 0 ?? =× jj ijk ??? ?=× jki ??? ?=× ikj ?? =× r 0 ?? =×kk 5. Variation of a vector 1 The Magnitude changes, the direction is preserved; 2 The direction changes, the magnitude is preserved; 3 Both the magnitude and direction change. i A r f A r A r ? AAA AAA if if rrr rrr ? ? += ?= Discuss 1 and 2 §2.3 the Cartesian representation of any vector t B t A BA t d d d d )( d d rr rr +=+ t B AB t A BA t d d d d )( d d r rr r rr ?+?=? t B AB t A BA t d d d d )( d d r rr r rr ×+×=× Differentiation of a vector: k t A j t A i t A kAjAiA tt A z y x zyx ? d d ? d d ? d d ) ??? ( d d d d ++=++= r 9 Homework 1 read chapter 2, especially the chapter summary; 2 problems: 3, 7, 15, 20, 30, 43, 51, 58 3 preparation of chapter 3