Catesian Coodinate System and Vectos
Coodinate System Coodinate system: used to descibe the position of a point in space and consists of 1. An oigin as the efeence point 2. A set of coodinate axes with scales and labels 3. Choice of positive diection fo each axis 4. Choice of unit vectos at each point in space Catesian Coodinate System
Vectos
Vecto A vecto is a quantity that has both diection and magnitude. Let a vecto be denoted by the symbol A The magnitude of A is denoted by A! A
Application of Vectos (1) Vectos can exist at any point P in space. (2) Vectos have diection and magnitude. (3) Vecto Equality: Any two vectos that have the same diection and magnitude ae equal no matte whee in space they ae located.
Vecto Addition Let A and B be two vectos. Define a new vecto C = A + B,the vecto addition of A and B by the geometic constuction shown in eithe figue
Summay: Vecto Popeties Addition of Vectos 1. Commutativity A + B = B + A 2. Associativity (A + B) + C = A + (B + C) 3. Identity Element fo Vecto Addition 0 such that A + 0 = 0 + A = A 4. Invese Element fo Vecto Addition!A such that A!A = 0 Scala Multiplication of Vectos + ( ) 1. Associative Law fo Scala Multiplication b( c A) = ( bc )A = (cb A) = c ( b A) 2. Distibutive Law fo Vecto Addition c (A + B) = c A + c B 3. Distibutive Law fo Scala Addition (b + c) A = b A + c A 4. Identity Element fo Scala Multiplication: numbe 1 such that 1 A = A
Vecto Decomposition Choose a coodinate system with an oigin and axes. We can decompose a vecto into component vectos along each coodinate axis, fo example along the x,y, and z-axes of a Catesian coodinate system. A vecto at P can be decomposed into the vecto sum, A = A + A + A x y z
Unit Vectos and Components The idea of multiplication by eal numbes allows us to define a set of unit vectos at each point in space ˆ ˆ ˆ (i, j, k ) ˆi = 1, ˆ j = 1, k ˆ = 1 with Components: A = (A, A, A ) A = A î, A = A ĵ, x x y y x y z A = A kˆ z z A = Ax î + A y ĵ + A z kˆ
Vecto Decomposition in Two Conside a vecto A = ( A, A, 0) Dimensions x- and y components: Magnitude: x y A x = A cos(!), A y = Asin(!) A = 2 2 A x + A y Diection: A y A x = Asin(!) Acos(!) = tan(!)! = tan "1 ( A y / A x )
Vecto Addition A = A cos(! A ) î + Asin(! A ) ĵ B = B cos(! B ) î + Bsin(! B ) ĵ Vecto Sum: C = A + B Components C x = A x + B x, C y = A y + B y C x = C cos(! C ) = A cos(! A )+ B cos(! B ) C y = C sin(! C ) = Asin(! A ) + Bsin(! B ) C = (Ax + B x ) î + (A + B ) ĵ = C cos(! ) î + C sin(! ) ĵ y y C C
Checkpoint Poblem: Vecto Decomposition Two hoizontal opes ae attached to a post that is stuck in the gound. The opes pull the post poducing the vecto foces A = 70 N î + 20 N ĵ and B =!30 N î + 40 N ĵ as shown in the figue. Find the diection and magnitude of the hoizontal component of a thid foce on the post that will make the vecto sum of foces on the post equal to zeo.
Checkpoint Poblem: Sinking Sailboat A Coast Guad ship is located 35 km away fom a checkpoint in a diection 42 0 noth of west. A distessed sailboat located in still wate 20 km fom the same checkpoint in a diection 36 0 south of east is about to sink. Daw a diagam indicating the position of both ships. In what diection and how fa must the Coast Guad ship tavel to each the sailboat?
Peview: Vecto Desciption of Motion Position ( t ) = x ( t ) ˆ i + y ( t ) ˆj Displacement! ( t ) =! x ( t ) ˆ i +! y ( t ) ˆj Velocity dx ( t ) ˆ dy ( t ) v ( t ) = i + ˆ j! v ( ) ˆ ( ) ˆ x t i + v y t j dt dt Acceleation dv ( ) ˆ ( ) x t dv y t a ( t ) = i + ˆ j! a ( ) ˆ ( ) ˆ x t i + a y t j dt dt
Dot Poduct A scala quantity Magnitude: A " B = A B cos! The dot poduct can be positive, zeo, o negative Two types of pojections: the dot poduct is the paallel component of one vecto with espect to the second vecto times the magnitude of the second vecto A " B = A (cos! ) B = A B A " B = A (cos! ) B = A B
Dot Poduct Popeties A! B = B! A c A! B = c ( A! B ) ( A + B )! C = A! C + B! C
Dot Poduct in Catesian Coodinates With unit vectos ˆ i, ˆ j and k ˆ ˆ i! ˆ i = ˆ j! ˆ j = k ˆ! k ˆ = 1 ˆ i! ˆ j = ˆ i! k ˆ = ˆj! k ˆ = 0 Example: ˆ i " ˆ i = ˆ i ˆi cos(0) = 1 ˆ i " ˆ j = ˆ i ˆj cos(! /2) = 0 A = A ˆ ˆ ˆ ˆ ˆ ˆ x i + A y j + A z k, B = B x i + B y j + B z k A! B = A B + A B + A B x x y y z z
Checkpoint Poblem: Scala Poduct In the methane molecule, CH4, each hydogen atom is at the cone of a tetahedon with the cabon atom at the cente. In a coodinate system centeed on the cabon atom, if the diection of one of the C--H bonds is descibed by the vecto A = î + and the diection of an adjacent C--H is descibed by the ĵ vecto + kˆ B = î! ĵ! k, ˆ what is the angle between these two bonds.
Summay: Coss Poduct Magnitude: equal to the aea of the paallelogam defined by the two vectos A # B = A B sin! = A B sin! = A sin! B (0 $! $ " ) ( ) ( ) Diection: detemined by the Right-Hand-Rule
Popeties of Coss Poducts A! B = " B! A c ( A! B ) = A! c B = c A! B ( A + B )! C = A! C + B! C
Coss Poduct of Unit Vectos Unit vectos in Catesian coodinates ˆ i " ˆ j = ˆ i ˆj sin! 2 = 1 ( ) ˆ i " ˆ i = ˆ i ˆj sin(0) = 0 ˆ i! ˆ j = k ˆ ˆ i! ˆi = 0 ˆ j! k ˆ = ˆ i ˆ j! ˆj = 0 k ˆ! ˆ i = ˆj k ˆ! k ˆ = 0
Components of Coss Poduct A = A ˆ i + A ˆ j + A k ˆ, B = B ˆ i + B ˆj + B k ˆ x y z x y z A! B = ( A B " A B ) ˆ i + ( A B " A B ) ˆ j + ( A B " A B ) k ˆ = y z z y z x x z x y y x ˆ i ˆ j k ˆ A A A x y z B B B x y z
Checkpoint Poblem: Vecto Poduct Find a unit vecto pependicula to A = î + ĵ! ˆk and B =!2î! ĵ + 3ˆk.
MIT OpenCouseWae http://ocw.mit.edu 8.01SC Physics I: Classical Mechanics Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems.