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 length of arrow magnitude of vector K A vector quantity is written in bold A or with a little arrow overhead A K A no arrow not bold A 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 Ay A x Ax G In 2D we need 2 numbers to specify a vector A magnitude A and angle or components Ax and Ay more on components later Addition of Vectors y B B C A G G G A B C A x PHYS1110 Notes Dubson 8 22 2009 University of Colorado at Boulder Vectors 2 G G G G Vector addition is commutative A B B A Graphical addition tip to tail or tail to head method A B C C 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 G G G G G G A B C D E S D E C B S A Definition of negative of vector same size opposite direction A A Definition of multiplication of a vector by a number b 3 A c 2 bA 3 times as long as A B cB negative c flips direction What about multiplication of a vector by a vector There are two different ways to define K K multiplication of two vectors K K 1 Dot product or scalar product A B and 2 Cross product A B These will be defined later PHYS1110 Notes Dubson 8 22 2009 University of Colorado at Boulder Vectors 3 Vector subtraction K K K K A B A B substract means add negative of K K K Graphically D A B B D B B A B A D A K K K D A B is the same as A D B K K K D B A is same as K K K B D A Components of a Vector K A A x x A y y y x x hat is the unit vector explained below A Ay x Ax 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 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 light rays y A x Ax y 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 B By Bx x PHYS1110 Notes Dubson 8 22 2009 University of Colorado at Boulder Vectors 4 Ax A x A cos A A sin y A y A sin A cos A Ay Ax A Ax 2 A y2 tan Ay Ax G Magnitude A A 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 x pronounced x hat is the unit vector pointing K in the x direction Similarly for y Do you see that A A x x A y y Note Ax is a number not a vector A x x is a vector because it is a number Ax times a vector x A x x A y y is the sum of two vectors y y x x Example of vector math Ax 2 Ay 3 What is the magnitude A and the angle with the x axis y x A A 3 tan Ax 2 A y2 3 2 22 32 4 9 13 3 6 3 tan 1 56 3o 2 2 PHYS1110 Notes Dubson 8 22 2009 University of Colorado at Boulder Vectors 5 Vector Addition by Components K K K C A B C x A x Bx Proof by diagram y C y A y By C B A x Bx Ax Cx Similarly subtraction by components K K K D A B D x A x Bx D y A y By 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 ycomponents 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 y ry y location of object x y r rx x 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 PHYS1110 Notes Dubson 8 22 2009 University of Colorado at Boulder x Vectors 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 K r K In 2D or 3D we define the velocity vector as v lim t 0 t 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 path of object y r1 x v y r r r2 v r r2 r2 r r v r1 r x x K r K has x and y components The component equations are The vector equation v lim t 0 t …
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