Cartesian Coordinate System and Vectors

Transcription

1 Catesian Coodinate System and Vectos

2 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

3 Vectos

4 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

5 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.

6 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

7 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

8 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

9 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

10 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 )

11 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

12 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.

13 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?

14 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

15 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

16 Dot Poduct Popeties A! B = B! A c A! B = c ( A! B ) ( A + B )! C = A! C + B! C

17 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

18 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.

19 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

20 Popeties of Coss Poducts A! B = " B! A c ( A! B ) = A! c B = c A! B ( A + B )! C = A! C + B! C

21 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

22 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

23 Checkpoint Poblem: Vecto Poduct Find a unit vecto pependicula to A = î + ĵ! ˆk and B =!2î! ĵ + 3ˆk.

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