Chapter 3: Vectors and Coordinate SystemsCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.SystemsCoordinate Systems• Used to describe the position of a point in space• Coordinate system consists of– a fixed reference point called the origin– specific axes with scales and labels–instructions on how to label a point relative to the origin Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.–instructions on how to label a point relative to the origin and the axesCartesian Coordinate System• Also called rectangular coordinate system•x-and y-axes intersect Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.•x-and y-axes intersect at the origin• Points are labeled (x,y)3Polar Coordinate System–Origin and reference line are noted– Point is distance r from the origin in the Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.the origin in the direction of angle θ, ccw from reference line– Points are labeled (r,θ)4Polar to Cartesian Coordinates•Based on forming a right triangle from r and θ•x= rcos θCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.•x= rcos θ• y = r sin θ5Cartesian to Polar Coordinates• r is the hypotenuse and θan angletanyxθ=Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.θmust be ccw from positive x axis for these equations to be valid2 2xr x y= +6Vectors and Scalars•A scalar quantity is completely specified by a single value with an appropriate unit and has no direction.•A vector quantityis completely described by a Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.•A vector quantityis completely described by a number and appropriate units plus a direction.Vector Notation•When handwritten, use an arrow: • When printed, will be in bold print: A• When dealing with just the magnitude of a vector in print, an italic letter will be used: A or |A| •The magnitude of the vector has physical unitsArCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.•The magnitude of the vector has physical units• The magnitude of a vector is always a positive numberAdding Vectors Graphically• Continue drawing the vectors “tip-to-tail”• The resultant is drawn from the origin of A to the end of the last vectorCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.end of the last vector• Measure the length of Rand its angle– Use the scale factor to convert length to actual magnitude9Subtracting Vectors•Special case of vector addition• If A – B, then use A+(-B)Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.A+(-B)• Continue with standard vector addition procedure10Multiplying or Dividing a Vector by a Scalar•The result of the multiplication or division is a vector• The magnitude of the vector is multiplied or divided by the scalar•If the scalar is positive, the direction of the result is Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.•If the scalar is positive, the direction of the result is the same as of the original vector• If the scalar is negative, the direction of the result is opposite that of the original vectorComponents of a Vector•A component is a part• It is useful to use rectangular componentsCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.components– These are the projections of the vector along the x- and y-axes12Vector Component Terminology• Ax and Ayare the component vectors of A– They are vectors and follow all the rules for vectors•Aand Aare scalars, and will be referred to as Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.•Axand Ayare scalars, and will be referred to as the components of AComponents of a Vector• The x-component of a vector is the projection along the x-axis•The y-component of a vector is the projection cosxA Aθ=Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.•The y-component of a vector is the projection along the y-axissinyA Aθ=Components of a Vector•The previous equations are valid only if θ is measured with respect to the x-axis• The components are the legs of the right triangle whose hypotenuse is ACopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.whose hypotenuse is A– May still have to find θ with respect to the positive x-axis2 2 1and tanyx yxAA A AAθ−= + =Unit Vectors•A unit vector is a dimensionless vector with a magnitude of exactly 1.• Unit vectors are used to specify a direction and have no other physical significanceCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.have no other physical significanceUnit Vectors, cont.•The symbolsrepresent unit vectorskˆand,jˆ,iˆCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.represent unit vectors• They form a set of mutually perpendicular vectors 17Unit Vectors in Vector Notation•The complete vector can be expressed asiˆjˆCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.ˆ ˆˆx y zA A A= + +A i j k18Adding Vectors Using Unit Vectors•Using R = A + B• Then()()ˆ ˆ ˆ ˆx y x yA A B B= + + +R i j i jCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.• and so Rx= Ax+ Bxand Ry= Ay+ By()()( )( )ˆ ˆx x y yx yA B A BR R= + + += +R i jR2 2 1tanyx yxRR R RRθ−= + =Trig Function Warning•The component equations (Ax= A cos θand Ay= Asin θ) apply only when the angle is measured with respect to the x-axis (preferably ccw from the positive x-axis).θCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.• The resultant angle (tan θ= Ay/ Ax) gives the angle with respect to the x-axis.Adding Vectors with Unit VectorsCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Adding Vectors Using Unit Vectors – Three Directions• Using R = A + B()()( )()( )ˆ ˆ ˆ ˆˆ ˆˆ ˆˆx y z x y zx x y y z zA A A B B BA B A B A B= + + + + += + + + + +R i j k i j kR i j kCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.• Rx= Ax+ Bx, Ry= Ay+ Byand Rz= Az+ Bzetc.()x y zR R R= + +R2 2 2 1tanxx y z xRR R R RRθ−= + + =Chapter 3.