Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Chapter 3: Vectors To introduce the concepts and notation for vectors: quantities that, in a single package, convey both magnitude and direction To be able to visualize vectors, and perform arithmetic operations upon them (addition and subtraction) To define and make use of unit vectors To become able to use this language of magnitude and direction, as contrasted with the usage of component language, especially in two dimensions To understand the paradigmatic vector: the displacement vector Chapter 3 Goals:
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Obvious language for vectors in two dimensions: magnitude and direction x y \A\ A magnitude of A: written |A| (some books say A) it is just the length of A |A| is always positive or 0 technically it is a scalar because it doesn’t change if you rotate your coordinates |A| carries the units of A direction of A: written A usually counterclockwise from x axis units of A are degrees or radians
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Polar coordinates in 2 dimensions (x,y) are cartesian coordinates two numbers needed to specify full information convention for is to start along x and swing ccw r hypotenuse x adjacent y opposite r : a scalar because coordinate system’s orientation does not affect r x and y : not, technically, scalars, but they are numbers
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Polar coordinates in 2 dimensions r hypotenuse x adjacent y opposite Upshot of this: 2d vector arithmetic looks very much like working on positions in polar coordinates
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Pros of magnitude-direction language we can write A = {|A|, } = {A, } lends itself to obvious pictures can easily be converted to compass (map) language: East x, and North y addition and subtraction of vectors is done pictorially accuracy of pictorial addition and subtraction is limited to human ability with protractor and ruler laws of sines and cosines needed to calculate: messy only convenient in 2d!! 3d requires the dreaded spherical trigonometry because there are 2 angles!! Cons of magnitude-direction language
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The displacement vector: we move ‘where’ into a higher dimension at last! initial position is r i final position is r f displacement is r:= r f – r i we are subtracting!! Ouch!! notice : the positions are origin-dependent, but the displacement is not!! x riri rfrf rr y the two position vectors are drawn in standard position, which means their tail is at the origin: makes sense the displacement vector is not drawn in s.p. if it behaves like the displacement vector, then it is a vector!! It is the ‘paradigmatic’ vector.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Triangle method for adding vectors A (and B) are two displacement vectors with B following A magnitude : |A| or A direction: an angle , which looks to be 0° we add to it vector B, in triangle method: tip-to-tail B’s angle looks to be about 60° A is added to B, to give resultant R = A + B lay down A and B with ruler and protractor; draw R R’s angle R looks to be about 40° with A in s.p., B is not, but R is in s.p. BB RR
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Some more facts about vectors to add more than two vectors, just generalize the triangle method, tip to tail style A + B = B + A: commutative (A + B) + C = A + (B + C): associative the negative of a vector is – A: same length as A but opposite direction [additive inverse] There is a zero vector (no length) vectors carry units; when added, units must be the same cA: also a vector (in opposite direction if c < 0) but ‘rescaled’ in length: |cA| = |c| |A|
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. How to subtract one vector from another A – B := A + (– B): to subtract, just add the additive inverse or: put both A and B tail-to-tail: then C = A – B has its tail at the tip of B (the one subtracted) and has its tip at the tip of A (the one added) in other words, we can see that we have added C to B in the additive ‘triangle’ way, to arrive at A Cseems to ‘start’ at B and end at A Example: the displacement vector is precisely the change in the position vector!! {show Active Figure AF_0306}
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Addition Example : Taking a Hike A hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower. Using ruler and protractor, lay out the two displacements A and B. Then, with the same tools, measure R.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. A different and more flexible language: component language (like polars) x y |A| A AxAx AyAy in this sketch, A is larger than 90 but you can deal… |A| is the hypotenuse; |A| ≥ 0 A x is the adjacent and here < 0 A y is the opposite and here > 0 “drop a perpendicular” from tip to either Cartesian axis you have made a right triangle draw vector with tail at origin: standard position we write A =
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Converting from magnitude-angle to components Aspects of component language addition (subtraction ) of vectors is simplicity itself: add (subtract) the components as numbers! components are numbers but technically not scalars since components are different in a rotated coordinate system {show Active Figure AF_0303}
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Revisiting the vector addition example Section 3.4
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Finishing the Example Section 3.4 Now we would work back from component language to magnitude-angle language: ‘net displacement is 41.3 km, at a bearing of 65.9° East of North’ R = {41.3 km, 24.1°}
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. A third language that uses components: unit vectors x y |A| A i is a vector of unit length, with no units, that points along x j and k are similar, along y (and z) create the vectors i A x & j A y j A y i A x A truly explicit way to write A remember: |i| = |j| = |k| = 1 one unit long, but carry no units, so the name ‘unit vector’ is dumb
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. What other operations can we do with vectors? cannot divide by a vector; vectors are only ‘upstairs’ dot (scalar (inner)) product of two cross (vector (outer (wedge))) product of two we can take their derivative with respect to a scalar we can integrate them but usually we integrate some kind of scalar product… The scalar product of two vectors A B put them tail-to-tail, with the angle between (0° ≤ ≤ 180°) the result is indeed a scalar A∙B = B ∙A (commutative)
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. More about the scalar product a measure of the parallelness of the vectors, as well as the magnitudes A∙B = (|A|)(|B| cos ) = length of A times the length of B’s projection along the line of A A∙B = (|B|)(|A| cos ) = length of B times the length of A’s projection along the line of B A∙A = |A| 2 i∙i = j∙j = k∙k =1 i∙j = j∙k = 0 etc. a vector’s component in a certain direction is the scalar product of that vector with a unit vector in that direction: C n = C∙n [to make a unit vector, just divide a vector by its magnitude: a = A/|A| ]