UNIVERSITY PHYSICS I CHAPTER 2 Introduction to Vector Analysis 82.1 scalar and vector quantities 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 otice:Not all things with a magnitude and direction are vectors
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
82.1 scalar and vector quantities 3. Representation of the vector arrow symbol magnitude A or A Ⅴ ector4 Letter symbol Direction 82.2 Operation of the vectors 1. Multiplication of a vector by a scalar >0 a<0 B B §22。 peration of the vectors 2. Vector addition by geometric methods Triangle rule or parallelogram rule: B Resultant: C=A+B Polygon rule R R=R1+R2+R3+R4 SNote:A+B≠A+B C≠A+B
2 §2.1 scalar and vector quantities Vector A r magnitude A or A r Direction 3. Representation of the vector §2.2 Operation of the vectors 1. Multiplication of a vector by a scalar B A r r =α α > 0 α < 0 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: R R1 R2 R3 R4 r r r r r = + + + R1 r R2 r R3 r R4 r R r §2.2 operation of the vectors A r B r C r B r C A B r r r Resultant: = + Note: C A B A B A B ≠ + + ≠ + r r
§22。 peration of the vectors 3. Vector difference by geometric methods The operation + ri is identical in all respects to the operation implied by C=A-B=A+(B N0eA-B≠B-AC≠A-B 4. The scalar(dot) product of two vectors The scalar product is a way to multiply two vectors to yield a scalar result Define: AB=ABcos A 82.2 operation of the vectors Notice OA, B are always positive ②60 ③60>π/2A·B= ACos<0 ④6=π/2A·B= ACos6=0 ⑤A(B+C)=A·B+AC 5. The cross product of two vectors The new vector that results from the vector product has both a magnitude and a direction Define: C=AxB
3 3. Vector difference by geometric methods The operation +(- ) is identical in all respects to the operation implied by - 1r r 1r r A r B r C A B A ( B) r r r r r = − = + − 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: A⋅ B = ABcosθ r r §2.2 operation of the vectors Note: A− B ≠ B − A C ≠ A− B r r r r 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: C A B r r r = × 1 A, B are always positive 2 θ π/2 4 θ = π/2 5 A⋅ B = ABcosθ > 0 r r A⋅ B = ABcosθ < 0 r r A⋅ B = ABcosθ = 0 r r A B C A B A C r r r r r s r ⋅( + ) = ⋅ + ⋅ Notice:
§22。 peration of the vectors Magnitude: AxB=ABsin 8 C=AxB Direction: perpendicular to the plan containing the A and B Right hand rule for the direction of the cross product B 密爱B A A 82.2 operation of the vectors o 0is always less than r in ABsin 0 ②A×A=00rB×B=0 ③AxB≠B×A ④A×B=-B×A ⑤A×(B+C)=AXB+A×C 82.3 The Cartesian representation of any vector 1. The Cartesian coordinate system Right handed Cartesian coordinate system
4 C A B r r r = × §2.2 operation of the vectors Magnitude: A× B = ABsinθ r r Direction:perpendicular to the plan containing the A B r r 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 1 θ is always less than π in ABsin θ 2 3 4 5 A× A = 0 or B× B = 0 r r r r A B B A r r r r × ≠ × A B B A r r r r × = − × A B C A B A C r r r r r r r ×( + ) = × + × §2.3 The Cartesian representation of any vector 1. The Cartesian coordinate system Right handed Cartesian coordinate system:
8 2.3 the Cartesian representation of any vector Unit(basis) vectors: i,j, k Magnitudes: 1 Directions: indicate the directions that correspond coordinates Increase Scalar product of the Cartesian unit vectors with each other =0.k 0 11=0 0k.=0k.k=1 8 2.3 the Cartesian representation of any vector 2. The case of two dimensions A A.i+4 Acos bi+asin e Ⅴ ector a gnitude:A=A2+A2 Direction: decided by angle e=tan A
5 §2.3 the Cartesian representation of any vector Unit(basis) vectors: i j k ˆ ,ˆ ,ˆ Magnitudes: 1 Directions: indicate the directions that correspond coordinates increase. i ˆ k ˆ j ˆ x y z 1 ˆ ˆ i ⋅ i = 0 ˆ ˆ i ⋅ j = 0 ˆ ˆ i ⋅ k = 0 ˆ ˆ j ⋅ i = 1 ˆ ˆ j ⋅ j = 0 ˆ ˆ j ⋅ k = 0 ˆ ˆ k ⋅ i = 0 ˆ ˆ k ⋅ j = 1 ˆ ˆ k ⋅ k = Scalar product of the Cartesian unit vectors with each other §2.3 the Cartesian representation of any vector 2. The case of two dimensions A i A j A A i A j x y ˆ sin ˆ cos ˆ ˆ = θ + θ = + r A r Ax Ay x y i ˆ j ˆ θ Vector A r Magnitude: 2 2 A = Ax + Ay Direction: decided by angle x y A 1 A tan− θ =
8 2.3 the Cartesian representation of any vector 3. The case of three dimensions A=Ai+Aj+Ak n prove that A=(4.4)2=(42+42+4 Ai=Acos%= A Ak A·j= Acos B Ak=AcOS y=A s 2.3 the Cartesian representation of any vector cos a+ cos B+cos y=1 4. The operations of vectors in Cartesian coordination system Addition: A+B=(Ai+A j+A, k)+(Bi+B j+B,k) (4+B3)+(A,+B)j+(42+B)k
6 §2.3 the Cartesian representation of any vector 3. The case of three dimensions A r A i x ˆ Azk ˆ A j y ˆ x y z A A i A j A k x y z ˆ ˆ ˆ = + + r z y x A k A A A j A A A i A A ⋅ = = ⋅ = = ⋅ = = γ β α cos ˆ cos ˆ cos ˆ r r r We can prove that 1 2 2 2 2 1 2 ( ) ( ) A A A = Ax + Ay + Az = ⋅ r r §2.3 the Cartesian representation of any vector 4. The operations of vectors in Cartesian coordination system Addition: A B i A B j A B k A B A i A j A k B i B j B k x x y y z x x y z x y z ˆ ( ) ˆ ( ) ˆ ( ) ) ˆ ˆ ˆ ) ( ˆ ˆ ˆ ( = + + + + + + = + + + + + r r cos cos cos 1 2 2 2 α + β + γ = x y z γ β α A r
8 2.3 the Cartesian representation of any vector Difference: A-B=(AI+A j+A, k)-B i+B j+B k) (A-B+(4-B)j+(4-B2)k Multiplication aA=a(Ai+Aj+A, k) =a4.i+a4,j+a4 Scalar product of two vectors: A B=(AI+A j+a, k). B i+Bj+B, k) AB+A, B,+Az 8 2.3 the Cartesian representation of any vector cross product of two vectors: AxB=(AI+A, j+A, k)(Bi+B,j+B, k) =(4,B-AB,)i+(4B-AB2)j +(AB1-1B)k i j k or AxB=A. A A B. BB Mnemonic Lijia
7 §2.3 the Cartesian representation of any vector A B i A B j A B k A B A i A j A k B i B j B k x x y y z z x y z x y z ˆ ( ) ˆ ( ) ˆ ( ) ) ˆ ˆ ˆ ) ( ˆ ˆ ˆ ( = − + − + − − = + + − + + r r Difference: Multiplication: A i A j A k A A i A j A k x y z x y z ˆ ˆ ˆ ) ˆ ˆ ˆ ( α α α α α = + + = + + r Scalar product of two vectors: x x y y z z x y z x y z A B A B A B A B A i A j A k B i B j B k = + + ⋅ = + + ⋅ + + ) ˆ ˆ ˆ ) ( ˆ ˆ ˆ ( r r §2.3 the Cartesian representation of any vector cross product of two vectors: A B A B k A B A B i A B A B j A B A i A j A k B i B j B k x y y x y z z y z x x z x y z x y z ˆ ( ) ˆ ( ) ˆ ( ) ) ˆ ˆ ˆ ) ( ˆ ˆ ˆ ( + − = − + − × = + + × + + r r x y z x y z B B B A A A i j k A B ˆ ˆ ˆ × = r r or Mnemonic: i j k i j k ˆ ˆ ˆ ˆ ˆ ˆ + -
82.3 the Cartesian representation of any vector where ixi=0 ixj=k ixk=-7 x7=-×1=0xk k×l= kxj=-1k×k=0 5. Variation of a vector O The Magnitude changes, the direction is preserved; 3 The direction changes, the magnitude is preserved; 3 Both the magnitude and direction change. 4A A Discuss①and② A1+ 8 2.3 the Cartesian representation of any vector Differentiation of a vector. da d dA d dA dt dt (AI+A, j+ A k) 十 十 da dB (4+B) 十 da B+A dB d da dB A×B) xB+Ax d 8
8 §2.3 the Cartesian representation of any vector where 0 ˆ ˆ i × i = j i k ˆ ˆ ˆ × = − k i j ˆ ˆ ˆ × = i j k ˆ ˆ ˆ × = 0 ˆ ˆ j × j = k j i ˆ ˆ ˆ × = − i k j ˆ ˆ ˆ × = − j k i ˆ ˆ × = r 0 ˆ ˆ k × k = 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. Ai r Af r A r ∆ A A A A A A f i f i r r r r r r ∆ ∆ = + = − Discuss 1 and 2 §2.3 the Cartesian representation of any vector t B t A A B t d d d d ( ) d d r r r r + = + t B B A t A A B t d d d d ( ) d d r r r r r r ⋅ = ⋅ + ⋅ t B B A t A A B t d d d d ( ) d d r r r r r r × = × + × Differentiation of a vector: k t A j t A i t A A i A j A k t t A x y z x y z ˆ d d ˆ d d ˆ d d ) ˆ ˆ ˆ ( d d d d = + + = + + r
Homework O read chapter 2, especially the chapter summary; ② problems:3,7,15,20,30,43,51,58 3 preparation of chapter 3
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
