Lecture 3 (Scalar and Vector Multiplication & 1D Motion) Physics 160-02 Spring 2017 Douglas Fields
Multiplication of Vectors OK, adding and subtracting vectors seemed fairly straightforward, but how would one multiply vectors? There are two ways to multiply vectors and they give different answers Dot (or Scalar) Product Cross (or Vector) Product You use the two ways for different purposes which will become clearer as you use them.
Dot (or Scalar) Product The dot product of two vectors is written as:. The result of a dot product is a scalar (no direction). There are two ways to find the dot product: A B A B cos AB AB cos or, A B Ax Bx Ay By Az Bz AB But what IS the dot product I mean what does it MEAN???
Dot (or Scalar) Product The dot product of two vectors gives the length of A in the direction of B (projection of A onto B ) times the length of. Example: y A Aj ˆ y Aiˆ x B Aiˆ Aj ˆ x A A iˆ A ˆj A Acos x x y A Asin y AB A iˆ A iˆ cos A 1cos Acos A ˆj A ˆj cos A 1cos 90 Asin
Dot (or Scalar) Product The dot product of two vectors gives the length of A in the direction of B times the length of B. Using the other method: AB y A iˆ A 1 A 0 A 0 A x y z x A ˆj A 0 A 1 A 0 A x y z y A kˆ A 0 A 0 A 1 A x y z z Aj ˆ y A Aiˆ x x A A iˆ A ˆj A kˆ x y z
Dot (or Scalar) Product The dot product of two vectors gives the length of A in the direction of B times the length of B. Another Example: AB A B A B cos Acos B A B AB B y A B A B A B B ; B A A B A x
Dot (or Scalar) Product Physics Example: Work force acting over a distance. W F D
CPS Question 3-1 Which of the dot products AB has the greatest absolute magnitude? A. B. C. y B B y y A x A B A x x D. y B A x
Dot (or Scalar) Product Commutative and Distributive Laws are obeyed by the dot product: A B B A A B C A B A C
Dot (or Scalar) Product This explains the second method then: ˆ ˆ ˆ ˆ ˆ ˆ x y z x y z A B A i A j A k B i B j B k A iˆ B iˆ A iˆ B ˆj A iˆ B kˆ x x x y x z A ˆj B iˆ A ˆj B ˆj A ˆj B kˆ y x y y y z A kˆ B iˆ A kˆ B ˆj A kˆ B kˆ z x z y z z A B A B A B x x y y z z
Dot (or Scalar) Product Usefulness of combining the two methods: A B A B A B A B A B cos cos AB x x y y z z AB A B A B A B x x y y z z AB
Cross (or Vector) Product The cross product of two vectors is written as: A B. The result of a vector product is a vector (has direction). To find the magnitude of a cross product: A B A B sin AB ABsin Its direction is perpendicular to both A and, and given by the Right-Hand-Rule: B
Cross (or Vector) Product Another way of finding A B: A B A ˆ ˆ ybz Az By i Az Bx Ax Bz j Ax By AyBx k Good way to remember using determinant: ˆ iˆ ˆj kˆ A B A A A x y z B B B x y z
Cross (or Vector) Product Commutative law is NOT obeyed by the cross product: Distributive law is obeyed: A B B A A B C A B A C
Cross (or Vector) Product Let s use this to get the second method: ˆ ˆ ˆ ˆ ˆ ˆ x y z x y z A B A i A j A k B i B j B k A iˆ B iˆ A iˆ B ˆj A iˆ B kˆ x x x y x z A ˆj B iˆ A ˆj B ˆj A ˆj B kˆ y x y y y z A kˆ B iˆ A kˆ B ˆj A kˆ B kˆ z x z y z z with iˆ ˆj kˆ ; ˆj kˆ iˆ ; kˆ iˆ ˆj then ˆ ˆ ˆ y z z y z x x z x y y x A B A B A B i A B A B j A B A B k
Cross (or Vector) Product But what IS the vector product I mean what does it MEAN??? It gives a sense of the perpendicularity and length of two vectors.
Cross (or Vector) Product Physics Example: Torque what does it take to turn a sticky bolt? r F
CPS Question 4-1 A B?, A, B in the x-y plane y A A. 0 45 o 45 o B x B. ABsin 45 o C. D. E. AB AB AB in the positive z-direction in the negative z-direction o sin 45 in the negative z-direction
CPS Question 4-2 What is A A B?, A, B in the x-y plane y A A. 0 B x B. 2 A B sin 45 o C. A 2 B in the positive z-direction D. A 2 B in the negative z-direction E. not enough information
Motion in One Dimension We need to define some terms: Distance (scalar) [m] Displacement (vector) [m] Speed (scalar) [m/s] Velocity (vector) [m/s] Acceleration (vector) [m/s 2 ] Time (scalar) [s]
y [cm] Motion and Graphs x [cm] 18 16 14 12 10 8 6 4 2 10s 9s 8s 7s 6s 0s 1s 2s 3s 4s 5s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 x [cm] 10 t [s]
Average Speed and Velocity When we talk about speed and velocity, we are referring to changes in distance and displacement over a change in time. Important to be specific about these changes: s avg total distance total time v avg total displacement x xf x total time t t t f i i
x [cm] 18 16 14 12 10 8 6 4 2 Average Speed and Velocity 1 2 3 4 5 6 7 8 9 What is the average speed from 0 5s? From 0 10s? t [s] 10 What is the average velocity from 0 5s? From 0 10s?
CPS Question 5-1 Someone walks 100m in what we will call the negative-x direction, then turns around and walks half way back, all at the same pace (speed). It takes 100s to walk the entire path. What is their average velocity? A. 1.5m/s B. 1.5m/s in the +x direction C. 0.5m/s D. 0.5m/s in the -x direction E. 1.0m/s in the -x direction
x [cm] Instantaneous Velocity 18 16 14 12 10 8 6 4 2 x 1 2 3 4 5 6 7 8 9 What is the average velocity from 1 2s? ˆ x x 6 2 ˆ f xi cm i cm i cm ˆ cm 1m ˆ m v ˆ avg 4 i 4 i 0.04 i t t t 2s 1s s s 100cm s f t i Note that this is the slope of the line connecting these two points. 10 t [s]
x [cm] Instantaneous Velocity 18 16 14 12 10 8 6 4 2 1 2 3 4 5 6 7 8 9 x dx v t lim vavg lim t 0 t 0 t dt What is the velocity at 2s? This is the definition of the derivative of the function that describes the position as a function of time. It gives a the slope of the tangent line to the function at any point. 10 t [s]
Instantaneous Velocity v [cm/s] 6 4 2-2 -4-6 1 2 3 4 5 6 7 8 9 10 t [s] We can now plot the velocity as a function of time. Notice that the velocity also changes. So, we can ask, what is the change in velocity as a function of time? This is known as the acceleration.
Exercise 2.10
Exercise 2.11
Instantaneous Acceleration v [cm/s] 6 4 2-2 -4-6 1 2 3 4 5 6 7 8 9 10 t [s] The acceleration is just given by the derivative of the velocity function. 2 a t v dv d dx d x lim aavg lim t 0 t 0 2 t dt dt dt dt So, it is also the second derivative of the position function.
Instantaneous Acceleration v [cm/s] 6 4 2-2 -4-6 1 2 3 4 5 6 7 8 9 10 t [s] a [cm/s 2 ] 6 4 2-2 -4-6 1 2 3 4 5 6 7 8 9 10 t [s]
Velocity and acceleration
Exercise 2.12
x [cm] 18 16 14 12 10 8 6 4 2 a [cm/s 2 ] 6 4 2 Curvature 1 2 3 4 5 6 7 8 9 10 t -2-4 -6 1 2 3 4 5 6 7 8 9 10 t [s]
Spherical Cows http://en.wikipedia.org/wiki/spherical_cow Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer "I have the solution, but it only works in the case of spherical cows in a vacuum."