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1 Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 37

2 3. Vectors Gols: To define vector components nd dd vectors. To introduce nd mnipulte unit vectors. To define nd understnd sclr product. To define nd understnd vector product. Pge 2 of 37

3 Vectors nd sclrs. Vectors: quntities which indicte both mgnitude nd direction. Exmples: displcemement, velocity, ccelertion Sclrs: quntities which indicte only mgnitude. Pge 3 of 37 Exmples: time, speed, mss

4 Vectors re represented by rrows: (i) The length of the rrow signifies mgnitude. (ii) The hed of the rrow signifies direction. Displcement vector for prticle trvelling from A to B on stright pth Note: All three vectors re identicl becuse they hve the sme direction nd mgnitude. Pge 4 of 37 A shift preserving both direction nd mgnitude does not chnge the vector. (Trnsltion.)

5 Displcement vector for prticle trvelling on curved pth. Note: independent of the pth from A to B. Pge 5 of 37 Nottion:, b, c,... or, b, c,... The mgnitude of vector : or

6 Adding vectors geometriclly Wht is the sum of two vectors? b + =? Step 1. Drw the vectors hed to til Pge 6 of 37 b

7 Step 2. The vector sum of nd b is the vector c pointing from the til of to the hed of b. b Mthemticl formul: + b = c c Pge 7 of 37

8 Commuttivity: + b = b + b b c Pge 8 of 37

9 Associtivity: ( + b ) + c = + ( b + c ) b c ( + b ) + c Pge 9 of 37 b c + ( b + c )

10 Inverse: + ( ) = 0 - Note: hs the sme mgnitude s, but it points in opposite direction Vector subtrction: b = + ( b ). Pge 10 of 37 -b - b b + b

11 Multiplying vectors by sclrs If vector, s 0 number then s = vector with mgnitude s = s Exmple nd direction 1.75 sme s if s > 0 opposite to if s < 0 Note: 0 = 0. Pge 11 of

12 i-clicker Which of the following sttements is flse for the three vectors below? A) + b + c = 0 b B) c + b = C) c < + b c D) c = + b E) None of the bove. Pge 12 of 37

13 Answer Which of the following sttements is flse for the three vectors below? A) + b + c = 0 c b B) c + b = C) c < + b D) c = + b E) None of the bove. Tringle inequlity: c < + b since, b, c not coliner. Pge 13 of 37

14 Components of vectors Axis = line equipped with preferred direction, lso clled orienttion. Exmple: one dimensionl motion positive direction x O = origin: x = 0 Pge 14 of 37

15 Projection: suppose nd given xis re in the sme plne 90 o 90 proj projection of long given xis o xis Note: proj is vector long the given xis. proj is not the component of long the given xis. (s stted in the textbook.) Pge 15 of 37

16 = = The component of long given xis is number = proj if proj points in the positive direction proj if proj points in the negtive direction Pge 16 of o xis 90 o xis 90 o proj = proj > 0 90 o proj = - proj < 0

17 = Right tringle rule = cos θ 90 o q proj = proj = xis cos q θ = ngle between the xis nd the vector (counterclockwise) Pge 17 of 37

18 = proj = - = cos ( q-p ) =- cos q 90 o proj proj q = cos q xis θ = ngle between the xis nd the vector (counterclockwise) Pge 18 of 37

19 Summry: The projection of is the vector proj. The component of is the number = cos θ Pge 19 of 37

20 Right hnded coordinte system: three mutully orthogonl xes meeting t point O clled origin. o 90 = p/ 2 z y 90 o = p/ 2 o 90 = p/ 2 O = origin x The x nd y xes re in the pge. The z-xis sticks out of the pge. x, y, z: coordintes Pge 20 of 37

21 The components of long the three xes y trnsltion , 2, 3 : the projections of on the x, y, z xes. x (vectors) Pge 21 of 37 z x, y, z : the components of long the x, y, z xes (numbers)

22 Plnr vectors in x, y plne Pge 22 of 37

23 The right tringle rules for plnr vectors x = cos θ y = sin θ = 2 x + 2 y Pge 23 of 37 tn θ = y x (if x 0).

24 Pge 24 of 37

25 i-clicker A vector is contined in the (y, z) plne such tht the ngle between nd the y xis is φ. Wht re the components of? z f y x A) x = cos φ, y = sin φ, z = 0 B) x = cos φ, y = 0, z = sin φ C) x = 0, y = sin φ, z = cos φ D) x = 0, y = cos φ, z = sin φ E) x = sin φ, y = 0, z = cos φ Pge 25 of 37

26 Answer A vector is contined in the (y, z) plne such tht the ngle between nd the y xis is φ. Wht re the components of? z f z y y x A) x = cos φ, y = sin φ, z = 0 B) x = cos φ, y = 0, z = sin φ C) x = 0, y = sin φ, z = cos φ D) x = 0, y = cos φ, z = sin φ E) x = sin φ, y = 0, z = cos φ Pge 26 of 37

27 Unit vectors Unit vector = vector of mgnitude 1 pointing in the positive direction long n xis xis u u =1 Pge 27 of 37

28 Unit vectors for right hnded coordinte system If hs components x, y, z, its projections re 1 = x î 2 = y ĵ 3 = z k Pge 28 of 37 = x î + y ĵ + z k

29 Two vectors re equl if nd only if their components re equl. = b x = b x, y = b y, z = b z. Pge 29 of 37

30 i-clicker Which of the following expressions is correct for the vector shown below? y A) = cosφî + sinφĵ B) = sinφî + cosφĵ f x C) = sinφî + cosφĵ D) = cosφî sinφĵ E) None of the bove. Pge 30 of 37

31 Answer Which of the following expressions is correct for the vector shown below? y A) = cosφî + sinφĵ B) = sinφî + cosφĵ C) = sinφî + cosφĵ Pge 31 of 37 x f y x D) = cosφî sinφĵ E) None of the bove.

32 Adding vectors by components For ny two vectors: = x î + y ĵ + z k b = bx î + b y ĵ + b z k we hve: + b = ( x + b x )î + ( y + b y )ĵ + ( z + b z ) k Pge 32 of 37 b = ( x b x )î + ( y b y )ĵ + ( z b z ) k More generlly, if s, t re sclrs, s + t b = (s x + tb x )î + (s y + tb y )ĵ + (s z + tb z ) k

33 Vectors nd the lws of physics Reltions mong vectors do not depend on the choice of coordinte system. Reltions in physics re lso independent of the choice of coordinte system. = 2 x + 2 y = ( x )2 + ( y )2 θ = θ + φ Pge 33 of 37

34 Sclr (dot) product Multiplying vectors Assocites to ny two vectors, b the number b = b cos φ = b commuttive Order is irrelevnt! Pge 34 of 37

35 Sclr product in unit vector nottion î î = ĵ ĵ = k k = 1 î ĵ = ĵ k = k î = 0 = x î + y ĵ + z k b = bx î + b y ĵ + b z k Pge 35 of 37 b = ( x î + y ĵ + z k) ( bx î + b y ĵ + b z k) = x b x + y b y + z b z

36 Vector (cross) product c to the plne of the two vectors direction of c: right hnd rule Pge 36 of 37 b = c c = b sin φ b = c Anti-commuttive Order is relevnt!

37 Vector product in unit vector nottion î ĵ = ĵ î = k ĵ k = k ĵ = î k î = î k = ĵ î î = ĵ ĵ = k k = 0 Pge 37 of 37 = x î + y ĵ + z k b = bx î + b y ĵ + b z k b = ( x î + y ĵ + z k) ( bx î + b y ĵ + b z k) = ( y b z b y z )î + ( z b x b z x )ĵ + ( x b y b x y ) k

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