PHY183-Lecture03 - V e c to rs ! Vectors are heavily used...

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Unformatted text preview: V e c to rs ! Vectors are heavily used in physics ! Need to manipulate them without any difficulty Physics for Scientists & Engineers 1 Fall Semester 2006 Lecture 3 ! Vector C • B e g in n in g a n d e n d p o in t • C h a r a c te r iz e d b y : M a g n itu d e D ir e c tio n U n it ! ! Remark: a quantity defined without a direction is a scalar August 29, 2006 Physics for Scientists&Engineers 1 1 August 29, 2006 Physics for Scientists&Engineers 1 2 C a r t e s ia n C o o r d in a t e s y s t e m ( 1 ) ! Used as a representation of vectors ! Quantifies a direction in 2-dimensional space T w o p e r p e n d ic u la r d ir e c tio n s C a r t e s ia n C o o r d in a t e s y s t e m ( 2 ) ! Quantifies a direction in 3-dimensional space T h ir d d ir e ct io n co m in g s t r aig h t o u t o f pag e x to th e r ig h t, y u p w a r d P o s itio n o f p o in t P s p e c ifie d b y H i g h e r d i m e n s i o n a li t i e s a r e u se d in m o d e r n t h e o r ie s ( b u t q u it e ab st r act an d h ar d t o r epr esen t on a tw od im e n sio n al pape r ) ( Px , Py ) ( Px , Py ) are positive or n e g a tiv e r e a l n u m b e r s August 29, 2006 Physics for Scientists&Engineers 1 3 August 29, 2006 Physics for Scientists&Engineers 1 4 R ig h t -H a n d R u le ! C a r t e s ia n c o o r d in a t e s y s t e m ( 3 ) ! A joins P and Q Q = ( 3,1) P = (!2, !3) ! A = (3 ! ( !2),1 ! ( !3)) = (5, 4) S h ift it to th e o r ig in fo r a s im p le r e p r e s e n ta tio n ! ! A joins R and S R = (!3, !1) S = (2, 3) Conventional assignment for right-handed coordinate system (more on 3D coordinate systems later in this semester) August 29, 2006 Physics for Scientists&Engineers 1 5 ! A = ( 2 ! ( !3), 3 ! ( !1) = (5, 4) August 29, 2006 Physics for Scientists&Engineers 1 6 ! ! !!! C = A+ B V e c t o r A d d it io n - G r a p h ic a l We learned: you can drag vectors around in space without changing their value • • Length stays the same Direction stays the same ! ! ! For every vector A there is a vector ! A, with the same length, pointing in the exact opposite ! ! direction ! !! A !A A + (! A) = 0 y ! Vector subtraction: !!! ! To obtain the vector!D = B ! A , ! we add the vector ! A to B , following the procedure of vector addition. V e c t o r S u b t r a c t io n ! In particular, you can drag vector B in such a way that its foot is at the tip of vector A Sum vector C then points from the foot of A to the tip of B You can do it the other way around ! !! AB ! ! !A !!!!! C = A+ B = B + A August 29, 2006 Physics for Scientists&Engineers 1 7 ! ! !x D = B! A 8 August 29, 2006 Physics for Scientists&Engineers 1 In V e c t o r S u b t r a c t io n O r d e r M a t t e r s ! Reverse the order and !! take! A ! B instead of ! B ! A . What is the result? ! The resulting vector !!! E = A! B is exactly the ! ! ! opposite vector to D = B ! A ! Rules for vector addition and subtraction are just like for real numbers. August 29, 2006 U n it V e c t o r s ! Vector representation in terms of unit vectors: y ! B ! !A ! ˆ ˆ ˆ A = ax x + a y y + az z ! 2D case ! ! !x D = B! A y ! !B y ! A x 9 !!! E = A! B Projection of A on the y axis provides its component ay ! ! A ˆ ax x ˆ ! x = (1, 0, 0) " ˆ # y = (0,1, 0) " z = (0, 0,1) $ˆ ˆ ay a y y ˆ y ˆ x ! ˆ ˆ A = ax x + a y y x 10 ax August 29, 2006 Physics for Scientists&Engineers 1 Physics for Scientists&Engineers 1 C o m p o n e n t M e t h o d fo r V e c t o r A d d it io n ! Vector addition can also be accomplished by using Cartesian components and unit vectors. ! Component representation ! ˆ ˆ ˆ A = ax x + a y y + az z ! ˆ ˆ ˆ B = bx x + by y + bz z !!! ! Vector addition C = A + B ˆ ˆ ˆ ˆ ˆ ˆ = [ax x + a y y + az z ] + [bx x + by y + bz z ] ˆ ˆ ˆ = (ax + bx ) x + (a y + by ) y + (az + bz ) z V e c t o r S u b t r a c t io n ! Procedure is exa! tly the same as vector addition c ! Difference vector: ! !! ˆ ˆ ˆ A = ax x + a y y + az z ! ˆ ˆ ˆ B = bx x + by y + bz z D = A! B ˆ ˆ ˆ ˆ ˆ ˆ = [ax x + a y y + az z ] ! [bx x + by y + bz z ] ˆ ˆ ˆ = (ax ! bx ) x + (a y ! by ) y + (az ! bz ) z ! With components: ! Components of sum vector ! ˆ ˆ ˆ C = cx x + c y y + cz z with cx = ax + bx cy = ay + by cz = az + bz ! ˆ ˆ ˆ D = dx x + d y y + dz z with d x = ax ! bx d y = ay ! by dz = az ! bz An equation between vectors equals three scalar equations! 11 August 29, 2006 Physics for Scientists&Engineers 1 12 August 29, 2006 Physics for Scientists&Engineers 1 ! V e c t o r le n g t h a n d d ir e c t io n ! A in component representation (in 2D) Vector ! ˆ ˆ A = ax x + a y y ! Calculation of its norm (=length) from its components y ˆ ay y ! A P Using Pythagoras in the right triangle OPQ ˆ y !Q ! A = ax 2 + a y 2 Also, the angle ! between A and the x axis O x ax x ˆ xˆ ! ! = arctan(a y / ax ) August 29, 2006 Physics for Scientists&Engineers 1 13 ...
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This note was uploaded on 03/01/2010 for the course PHY 183 taught by Professor Wolf during the Spring '08 term at Michigan State University.

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