PHY 2140 Alexey A PetrovVectorsA vector has both a magnitude and a direction. A scalar has only a magnitude, nodirection. Vectors are indicated by an arrow over the symbol, e.g. the velocity vector is written asvr. Vectors are represented by arrows.Length of arrow = magnitude of vector. The negative of avector br is called b-r and has the same magnitude as br and the opposite direction.Examples:Vectors: acceleration, velocity, displacement, force, electricfield.Scalars: speed, distance traveled, time, electric potential A vector is specified by its magnitude and direction - not byits starting point. The arrows in the picture on the right allrepresent the same vector!Adding VectorsExample: You hike 3 km north from your car, and then change direction and hike 2 km east. How far away from your car are you then?We need a special procedure to add vectors, so that the directions can be taken into account. There are two of them. First, a graphic method (or ”Tail-to-Tip”)1. draw vector ar with the correct size and angle2. draw vector br with the correct size andangle, so that the tail of br starts at the headof ar3. the sum of vectors s a b= +r r r is then found byconnecting the tail of ar with the head of brSecond method involves computation of componentsof vectors. Recall that any vector in the xy-plane canbe viewed as sum of two components: x yv v v= +r uur uurxvuur: is a vector along the x-axisyvuur: is a vector along the y-axisxvuur and yvuur form a right triangleFinding the components, magnitude and direction of a vector using trigonometryFrom trigonometry we know thatsin , cosyxvvv vJ J= =, so we can find:1. Components: sin , cosy xv v v vJ J= =2. Magnitude: 2 2x yv v v= +r3. Direction: tanyxvvJ =Adding two vectors by components1. Resolve both vectors into their components (with respect to the same axes), i.e. 1v with 1 1,x yv v and 2v with 2 2,x yv v2. Add the x components and the y components separately. This gives you the x and y components of the resultant vector, i.e. , 1 2sum x x xv v v= + and, 1 2sum y y yv v v= +.3. Follow instructions above on how to find the magnitude and direction of the resultant vectorNote: never add y and x
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