Scalars and Vectors Scalars and Vectors A scalar is a number which expresses quantity. Scalars may or may not have units associated with them. Examples: mass, volume, energy, money A vector is a quantity which has both magnitude and direction. The magnitude of a vector is a scalar. Examples: Displacement, velocity, acceleration, electric field 1 Vector Notation • Vectors are denoted as a symbol with an arrow over the top: xr • Vectors can be written as a magnitude and direction: r E = 15.7 N C@30o deg Vector Representation • Vectors are represented by an arrow pointing in the direction of the vector. • The length of the vector represents the magnitude of the vector. • WARNING!!! The length of the arrow does not necessarily represent a length. r A = 2.3m s 2 Vector Addition Adding Vectors Graphically. r r r r r A A B A B r r r r C = A + B B Arrange the The resultant is drawn vectors in a head from the tail of the first to tail fashion. to the head of the last vector. Vector Addition This works for any number of vectors. r B r r A C r r r r r r R = A + B + C + D D 3 Vector Addition Vector Subtraction Subtracting Vectors Graphically. r − B r B r r A A r r B B r r r r r C = A − B = A + (− B) Flip one vector. The resultant is drawn Then proceed to from the tail of the first add the vectors to the head of the last vector. 4 Vector Components Any vector can be broken down into components along the x and y axes. Example:r = 5.0m@30o from the horizontal. Find its components. rr = rr + rr x y r o ˆ rx = (5.0m)cos30 i r r r = 4.3miˆ r = r sinθˆj x θ y r o ˆ r ˆ ry = (5.0m)sin 30 j rx = r cosθi r ˆ ry = 2.5mj Vector Addition by Components You can add two vectors by adding the components of the vector along each direction. Note that you can only add components which lie along the same direction. r A = 3.2m siˆ + 2.5m s ˆj r + B = 1.5m siˆ + 5.2m s ˆj r r A + B = 4.7m siˆ + 7.7m s ˆj r r Never add the x-component A + B = 12.4 m s and the y-component 5 Unit Vectors Unit vectors have a magnitude of 1. They only give the direction. A displacement of 5 m in the x-direction is written as y r d = 5miˆ The magnitude is 5m. ˆj The direction is the î-direction. iˆ x kˆ z Finding the Magnitude and Direction Pythagorean Theorem 2 2 ry r = rx + ry tanθ = rx r r r ry θ −1 ry r θ = tan rx rx 6 Vector Multiplication I: The Dot Product The result of a dot product of two vectors is a scalar! r r A⋅ B = AB cosθ r A ˆ ˆ θ iˆ ⋅iˆ = 1 i ⋅ j = 0 r B ˆj ⋅ ˆj = 1 ˆj ⋅ kˆ = 0 kˆ ⋅ kˆ = 1 iˆ ⋅ kˆ = 0 Vector Multiplication I: The Dot Product r F = (2iˆ + 3ˆj − 2kˆ)N sr = (3iˆ − 4 ˆj − 6kˆ)m r F ⋅ sr = 2(3)N ⋅ m + 3(−4)N ⋅ m + (−2)(−6)N ⋅ m r F ⋅ sr = 6N ⋅ m 7 Vector Multiplication II: The Cross Product The result of a cross product of two vectors is a new vector! r r A× B = ABsinθ r A ˆ ˆ ˆ ˆ ˆ θ i ×i = 0 i × j = k r r C B ˆj × ˆj = 0 ˆj × kˆ = iˆ kˆ × kˆ = 0 kˆ ×iˆ = ˆj Vector Multiplication II: The Cross Product r r r q(vr × B)= (qvr × B)= (vr × qB) r r r r r r C = B × A = − A× B A ( ) ( ) θ r r r r r r C ⊥ A C ⊥ B C B 8 Vector Multiplication II: The Cross Product r F = (2iˆ + 3ˆj − 2kˆ)N r = (3iˆ − 4 ˆj − 6kˆ)m r τv = (r × F ) τv = ()()2N 3m (iˆ ×iˆ)+ (2N)(− 4m)(iˆ × ˆj)+ (2N)(− 6m)(iˆ × kˆ) + ()()3N 3m (ˆj ×iˆ)+ (3N)(− 4m)(ˆj × ˆj)+ (3N)(− 6m)(ˆj × kˆ) + ()()− 2N 3m (kˆ ×iˆ)+ (− 2N)(− 4m)(kˆ × ˆj)+ (− 2N)(− 6m)(kˆ × kˆ) τv = (26iˆ − 6 ˆj +17kˆ)N ⋅m Vector Multiplication II: The Cross Product r r F = (2iˆ + 3ˆj − 2kˆ)N r = (3iˆ − 4 ˆj − 6kˆ)m r τv = (r × F ) ˆj kˆ − 4 − 6 τv = (− 4(− 2)− (3)(− 6))iˆ 3 − 2 iˆ ˆj + (− 6(2)− (3)(− 2))ˆj 3 − 4 ˆ ˆ + (3(3)− (2)(− 4))kˆ k i 2 3 − 6 3 − 2 2 τv = (26iˆ − 6 ˆj +17kˆ)N ⋅m 9 Vector Multiplication II: Right Hand Rule Index finger in the direction of the first vector. Middle finger in the direction of the second vector Thumb points in the direction of the cross product. WARNING: Make sure you are using your right hand!!! 10.