Vectors-1Vectors A vector is a mathematical object consisting of a magnitude (size) and a direction.A vector can be represented graphically by an arrow: A vector quantity is written in bold (A) or with a little arrow overhead (Av)A (no arrow, not bold) = Av = magnitude of the vector = positive number (magnitudes are positive by definition)Examples of vector quantities: position, velocity, acceleration, force, electric field.If two vectors have the same direction and the same magnitude, then they are the same vectorVector = magnitude + direction (not location)In 2D, we need 2 numbers to specify a vector Ar: - magnitude A and angle ----or- components Ax and Ay (more on components later)Addition of Vectors A B C+ =r rr1/14/2019 © University of Colorado at BoulderyxAAAyx-AAxAyyxBAABClength of arrow = magnitude of vectordirection of arrow = direction of vectorVectors-2Vector addition is commutative: A B B A+ = +r rr rGraphical addition: "tip-to-tail" or "tail-to-head" method:Addition by "parallelogram method" (same result as tip-to-tail method)Can add lots of vectors (like steps in a treasure map: "take 20 steps east,then 15 steps northwest, then…")A B C D E S+ + + + =r r rr r rDefinition of negative of vector (same size, oppositedirection):Definition of multiplication of a vector by a number:What about multiplication of a vector by a vector? There are two different ways to define multiplication of two vectors: (1) Dot product or scalar product A B�vv and (2) Cross product A B�vvThese will be defined later.1/14/2019 © University of Colorado at BoulderABCBACABCABCDESAAb =3Ab A3 times as long as Ac = -2Bc Bnegative c flips directionVectors-3Vector subtraction:A B A ( B)- = + -v vv v- "substract" means "add negative of"Graphically: D A B= -vv vD A B is the same as D B A is same as B D A= - + = + =v v vv v v v v vComponents of a Vectorx yˆ ˆA A x A y= +v (ˆx= "x-hat" is the unit vector, explained below)Ax = A cos - = x-component = "projection of A onto x-axis"Ay = A sin -- = y-component = "projection of A onto y-axis"-Think of the Ax as the "shadow" or "projection" of thevector A cast onto the x-axis by a distant light sourcedirectly "overhead" in the direction of +y.Components are numbers, not vectors. They do not have adirection, but they do have a sign, a (+) or (–) sign. If the"shadow" onto the x-axis points in the +x direction, then Axis positive.Here, Bx is negative, because the x-projection is along the -x direction.By is positive, because the y-projection is along the +y direction.1/14/2019 © University of Colorado at BoulderAyx-AxAyByxBxByABBABBDDABDAAyx-Axlight raysVectors-4xxAcos A A cosAq = � = qyyAsin A A sinAq = � = q2 2x yA A A= +yxAtanAq =Magnitude A | A |=r is positive always, but Ax and Ay can be + or - .Example of vector math: Ax = +2, Ay = -3 What is the magnitude A, and the angle --with the x-axis-?2 2 2 2x yA A A 2 3 4 9 13 3.6= + = + = + = ;1 o3 3tan tan 56.32 2-����a = � a = =������1/14/2019 © University of Colorado at BoulderA-AxAyyx-A32Vectors-5Vector Addition by Components:x x xy y yC A BC A BC A B= + �= += +v vvSimilarly, subtraction by components:x x xy y yD A B D A BD A B= - � = -= -vv vPosition, Velocity, and Acceleration VectorsVelocity is a vector quantity; it has a magnitude, called the speed, and a direction, which is the direction of motion. Position is also a vector quantity. Huh? What do we mean by the magnitude and direction of position? How can position have a direction? In order to specify the position of something, we must giveits location in some coordinate system, that is, its locationrelative to some origin. We define the position vector r asthe vector which stretches from the origin of our coordinatesystem to the location of the object. The x- and y-components of the position vector are simply the x and ycoordinates of the position. Notice that that the positionvector depends on the coordinate system that we havechosen.If the object is moving, the position vector is a function of time r = r(t). Consider the position vector at two different times t1 and t2, separated by a short time interval -t = t2 – t1. (-t is read "delta-t") The position vector is initially r1, and a short time later it is r2. The change in positionduring the interval -t is the vector -r = r2 – r1. Notice that, although r1 and r2 depend on the 1/14/2019 © University of Colorado at BoulderyxABCAxBxCxProof by diagram:yxry = yrrx = xlocation of object (x , y)Vectors-6choice of the origin, the change in position -r = r2 – r1 is independent of choice of origin. Also, notice that change in something = final something – initial something. In 2D or 3D, we define the velocity vector as t 0rv limtD �D=Dvv.As -t gets smaller and smaller, r2 is getting closer and closer to r1,and -r is becoming tangent to the path of the object. Note thatthe velocity v is in the same direction as the infinitesimal -r ,since the vector v is a positive number (1/-t) times the vector -r.Therefore, the velocity vector, like the infinitesimal -r, is alwaystangent to the trajectory of the object.The vector equation t 0rv limtD �D=Dvv has x- and y-components. The component equations arex yt 0 t 0x yv lim , v limt tD � D �D D= =D D. Any vector equation, like A B C= +v vv , is short-handnotation for 2 or 3 component equations: x x xA B C= +, y y yA B C= +, z z zA B C= +The change in velocity between two times t1 and t2 is -v = v2 – v1 (remember that change is always final minus initial). We define the acceleration vector as t 0va limtD �D=Dvv. As we mentioned in the chapter on 1D motion, the direction of the acceleration is the same as the direction of -v. The direction of the acceleration is NOT the direction of the velocity, it is the “direction towards which the velocity is tending”, that is, the direction of -v. We will get more experience thinking about the velocity and acceleration vectors in the next few chapters.1/14/2019 © University of Colorado at Boulderyxr1r2-ryxr1r2-rr2path of object
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