Vectors-1 Vectors A vector is a mathematical object consisting of a magnitude (size) and a direction. A vector can be represented graphically by an arrow: direction of arrow = direction of vector A vector quantity is written in bold (A) or with a little arrow overhead () AKlength of arrow = magnitude of vectorA (no arrow, not bold) = AK = 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 vector y A Vector = magnitude + direction (not location) A A x y In 2D, we need 2 numbers to specify a vector AG: • magnitude A and angle θ or • components Ax and Ay (more on components later) Addition of Vectors AB C+=GGG x θ A AyAxy BB CA Ax PHYS1110 Notes, Dubson 8/22/2009 © University of Colorado at BoulderVectors-2 Vector addition is commutative: AB BA+=+GGGG A BC CGraphical addition: "tip-to-tail" or "tail-to-head" method: B A Addition by "parallelogram method" (same result as tip-to-tail method) C B A Can add lots of vectors (like steps in a treasure map: "take 20 steps east, then 15 steps northwest, then…") ABCDE S++++ =GGGGGG D E C B S A Definition of negative of vector (same size, opposite direction): 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⋅KK and (2) Cross product A B×KK These will be defined later. A−Ab =3 A b A 3 times as long as A c=−2c B negative c flips direction BPHYS1110 Notes, Dubson 8/22/2009 © University of Colorado at BoulderVectors-3 Vector subtraction: AB A(B)− =+−KKKK ← "substract" means "add negative of" Graphically: DA= −KKK BKyˆyhe x-projection is along the −x −BBA B−B D A B is the same as D B A is same as B D A= − += + =KKKK KK KK Components of a Vector xˆAAxA=+K (= "x-hat" is the unit vector, explained below) ˆx Ax = A cos θ = x-component = "projection of A onto x-axis" Ay = A sin θ = y-component = "projection of A onto y-axis" Think of the Ax as the "shadow" or "projection" of the vector A cast onto the x-axis by a distant light source directly "overhead" in the direction of +y. Components are numbers, not vectors. They do not have a direction, but they do have a sign, a (+) or (–) sign. If the "shadow" onto the x-axis points in the +x direction, then Ax is positive. Here, Bx is negative, because tdirection. By is positive, because the y-projection is along the +y direction. A y x θ Ax Ay B y x Bx By A D DABAD A y xθ light rays Ax PHYS1110 Notes, Dubson 8/22/2009 © University of Colorado at BoulderVectors-4 xxAcos A AcosAθ = ⇒ = θ A yyAsin A A sinAθ = ⇒ = θ 2xAAA=+2y yxAtanAθ = Magnitude A||A=G is positive always, but Ax and Ay can be + or − . Unit Vectors A unit vector is a vector with length one. Unit vectors are denoted with an "hat" or carrot symbol overhead ( ^ ). The unit vector (pronounced "x-hat") is the unit vector pointing in the +x direction. Similarly for . Do you see that ˆxˆyxyˆˆAAxA=+yK ? Note: Ax is a number, not a vector. is a vector, because it is a number (AxˆA xx) times a vector ( ). ˆx is the sum of two vectors xˆAx Ay+yˆ Example of vector math: Ax = +2, Ay = −3 What is the magnitude A, and the angle α with the x-axis ? 22 22xyA A A 2 3 4 9 13 3.6=+=+=+= 1o33tan tan 56.322−⎛⎞⎟⎜α = ⇒α==⎟⎜⎟⎜⎝⎠ θ Ay Ax y y^ ^ x xy x α A 3 2 PHYS1110 Notes, Dubson 8/22/2009 © University of Colorado at BoulderVectors-5 Vector Addition by Components: Proof by diagram: xxyyCABCABCAB=+⇒=+=+KKKxyxyB y Similarly, subtraction by components: xxyyDAB D ABDA= −⇒ = −= −KKK Position, Velocity, and Acceleration Vectors Velocity 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 give its location in some coordinate system, that is, its location relative to some origin. We define the position vector r as the vector which stretches from the origin of our coordinate system to the location of the object. The x- and y-components of the position vector are simply the x and y coordinates of the position. Notice that that the position vector depends on the coordinate system that we have chosen. 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 position during the interval ∆t is the vector ∆r = r2 – r1. Notice that, although r1 and r2 depend x A C B Bx Ax Cxy x ry = y r location of object (x , y) rx = x PHYS1110 Notes, Dubson 8/22/2009 © University of Colorado at BoulderVectors-6 on the choice 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 t0rvlimt∆→∆=∆KK. 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 that the 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 always tangent to the trajectory of the object. y he vector equation T t0rvlimt∆→∆=∆KK has x- and y-components. The component equations are xyt0 t0xyvlim ,vlimt∆→ ∆→∆∆==∆ t∆. Any vector equation, like C , is short-hand notation for 2 or 3 component equations: ABC=+ABC=+zChe change in velocity between two times t1 and t2 is ∆v = v2 – v1 (remember that change is AB=+KKKxxx, yyy, zzAB=+ Talways final minus initial). We define the acceleration vector as t0valim∆=KK. As we
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