Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Transcription

1 Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section find the vecto defined by two points and detemine the nom of the vecto. add two vectos multiply a non-zeo vecto by a non-zeo scala. epesent a non-zeo vecto in the xy-plane in tems of its magnitude and the angle it makes with the positive x-axis.

2 Vectos in D space Vecto algeba and vecto calculus have esulted fom pactical engineeing applications: Mechanics, Fluid flows, Wieless Communications Scala: is descibed by a single quantity such as wok, enegy, potential, speed, tempeatue, blood pessue.. Vecto: is descibed by a magnitude and diection such as velocity, electic foce, position of a obot Thee ae many quantities that ae vecto functions: Some Daily Use of Vectos A wind of 80 km/h fom the Southeast. A ca going 80 km/h East. A vetical velocity of 0 m/s. A plane taveling 000 km/h on a 80 heading. These issues ae descibed by a magnitude and a diection.

3 Some Applications of Vectos Mechanics: Foce, Toque, position, speed, acceleation, Electomagnetism: Electic and magnetic fields, cuent density, pointing vecto, Example Walking and Diffeent Foces Example Mechanical System in Equilibium Othe Examples of vecto quantities Notation v, u AB Acknowledgment: Most figues included in class notes ae copied fom the textbook by Zill and Cullen. 3

4 Notation and Teminology A vecto with stating point A and end point B is witten as Magnitude of AB is witten as: AB Example: In D Catesian Cood.: a a iˆ + a Magnitude : ˆj < a a, a a > [ a + a, a ] AB Two vectos with the same magnitude and diection ae equal Paallel vectos: nonzeo scala multiples of each othe 4

5 A note about notation The textbook uses boldface to epesent vectos, I may place an aow above geneal vectos and a hat ove unit vectos. I would like you all to clealy identify vectos in you wok. F F u û i iˆ 5

6 Addition of Vectos Conside two vectos AB and with common initial point A AC The sum of two vectos is the main diagonal of the paallelogam with the vectos as sides AD AB+ AC Example: (4i + 4 j) + (6i 6 j) 0i j 4i + 6 j + 6i 6 j 0i 0 j 0i 6

7 Subtaction Subtaction: The diffeence of AB AC AB+ ( AC) AB AC and is defined by AB AC is the main diagonal of the paallelogam with sides and AB AC O CB AB AC is a vecto fom the end of the second vecto towad the end of the fist vecto 7

8 Review Execise (page 346): Pob. 48 Find the magnitude of F and F. At equilibium: F + F + w 0 w -50j lb Sphee weight50 lb suppoting planes F 5.9 lb, F 36.6 lb 8

9 Popeties of Vectos Magnitude, length, o nom of a vecto a: a If a < a > then:, a a a + a A vecto that has magnitude is called unit vecto. A unit vecto in the diection of a is: uˆ a with u a j u The i, j unit vectos: i<,0>, j<0,> i < a i j, a > < a,0 > + < 0, a > a + a Example: Given a<3,-4>, fom a unit vecto in the same diection as a. Answe: <0.6,-0.8> In the opposite diection of a. Answe: <-0.6,0.8> 9

10 7. Vectos in 3-Space Rectangula o Catesian Coodinate D-Space: Two othogonal axes 3D-Space: Thee mutually othogonal axes The thee axes follow the Right Hand Rule 0

11 Coodinate Plane: Each pai of coodinate axes detemines a coodinate plane (xy,xz and yz). Octant:The coodinate planes divide the 3-space into 8 pats known as Octants. Fist octant: x, y, z>0 3D-Space

12 Position Vecto: Fo a point P, the position vecto is OP < x, y z, Distance Fomula between two points: Given points: > P ( x, y, z) & P ( x, y, z) d ( P,P ) ( x x) + ( y y) + ( z z ) P P Vecto between two points: PP < OP x OP x, y y, z z > Examples: P (,,3) & P (,-,-)

13 Component Definitions in 3D-Space Let a < a, a, a3 > and b, b, > be vectos in (i) Addition: a (ii) Scala Multiplication: (iii) Equality: if an only if (iv) Negative of a vecto: (v) Subtaction: (vi) Zeo vecto: 0 <0,0,0> (vii) Magnitude: b 3 < b3 + b < a + b, a + b, a3 + b3 > k a < ka, ka, ka3 > a b a b, a b, a3 b3 b < b, b, > b3 a b a + ( b) < a b, a b, a3 b3 a a + + a a3 R > a < a, a, a3 > ai + aj+ a3k 3

14 Unit Vectos in 3D space: i <,0,0>, j <0,,0>, k <0,0,> a < a, a, a3 > ai + aj+ a3k Example: If a 3i - 4j + 8k and b I - 4k, find 5a - b. b i- 0j - 4k b i + 0j - 8k 5a 5i - 0j + 40k 5a - b 3i - 0j + 48k 4

15 7.3 Dot (scala o inne) Poduct. Section 7.3 define the dot (inne) poduct (a. b) and intepet it geometically. use the dot poduct to detemine: wok done by a foce, the angle between two vectos, whethe two vectos ae pependicula to one anothe, pojections and components of vectos, diection angles and diection cosines 5

16 Dot (scala o inne) Poduct Applications: Mechanics and Electomagnetism Definition: The dot poduct of two vectos a and b is the scala a b a b cosθ & 0 θ π θ is the angle between a and b Example Dot poduct i. i, j. j, k. k since i j k and θ 0 i. j0, j. k0, k. i0 since θ 90 o Example Given: a0i+j-6k, b-0.5i+4j-3k a. b(0)(-0.5)+()(4)+(-6)(-3) 6

17 Physical Intepetation of the Dot Poduct A constant foce of magnitude F moves an object a distance d in the diection foce, the wok done by the foce (W): W F d ( F F W cosθ ) d d cosθ F d F d When a constant foce F applied to a body acts at an angle θ to the diection of motion, the wok done by the foce (W): Note: if F and d ae othogonal, W0. Examples: 7

18 P7.3-47: Given F 30N d < 4,3 > m weight Wok done by w (gavity foce) w. d 0 (w d) Wok done by F (applied foce) F. d F. d cos θ (d // F) 50 N.m 8

19 Popeties of Dot Poduct a. b 0 if a0 o b0 a. b b. a (commutative law) a. (b+c) a. b+a. c (distibutive law) a. (kb) (ka). b k(a. b) k a scala a. a 0 a. a a Fo nonzeo vectos a and b (i) a. b > 0 if and only if θ is acute (ii) a. b < 0 if and only if θ is obtuse, and (iii) a. b 0 if and only of if cos θ 0 (Othogonal vectos) Theoem 7. Citeion fo Othogonal Vectos Two nonzeo vectos a and b ae othogonal if and only if a. b0 9

20 Angle Between Two Vectos: cos a b ab + ab + a3b θ 3 a b a b Example Find the angle between a i+3j+k & b -i+5j+k. Solution: a. b4, a 4, b 7 cos θ 4 7 θ cos o

21 Fo a nonzeo vecto in 3D-Space the angles α, β and γ with i, j, and k ae called diection angles of a. a a i + aj+ a3k a iˆ a cosα a iˆ a Diection cosines fo Diection Angles a cos β a cos γ a 3 a cos α + cos β + cos γ Example Find the diection cosines and diection angles of the vecto a i+5j+4k. a 6.7 α o o 7.7, β 4.8, γ 53.4 o

22 Component of a on b: comp b a a b a cos ˆ θ a b b scala Pojection of a in the diection of b: poj a (comp a b ) ( ˆ ) ˆ b b a b b b vecto Example: a < -,-,7 > & b < 6,-3,- >

23 7.4 Coss (Vecto) Poduct 3. Section 7.4 define the vecto (coss) poduct (a x b) and intepet it geometically. detemine the coss poduct of vectos and combinations of vectos, use to detemine toque find unit vectos that ae pependicula to two given vectos. 3

24 Coss (Vecto) Poduct The vecto poduct of vectos A and B is given by ˆi ˆj kˆ A B A B A B x x A B y y A B z z iˆ( A A B sin( θ ) nˆ B aea of paalleogam y z A B z y ) + kˆ( A B x y A ˆ( j A B y B x x ) z A B z n A x ) θ AxB B whee n is a unit vecto pependicula to A and B, pointing in the diection given by the ight hand scew ule (i.e. the diection in which a scew would advance if it wee tuned fom A to B. Example: a < -,-,7 > & b < 6,-3,- > 4

25 Typical Applications a) AREA OF A PARALLELOGRAM with edges a and b: Aea a b a b sin θ b) AREA OF A Tiangle with edges a and b: Aea / a b / a b sin θ Example (p7.4, # 48): Find aea of the tiangle though: p (0,0,0), p (0,,), P3 (,,0) 5

26 Typical Applications Volume of a paallelepiped (with edges: a, b & c) Volume (aea of base). (height) b c. comp b c a b c. a (b c) / b c Volume a (b c) Example: a < 3,, >, b <,4, > & c <,,5 > 6

27 MOMENT OF A FORCE Typical applications In mechanics the moment m of a foce F about a point Q is defined as the poduct m F d whee d is the (pependicula) distance between Q and the line of action L on F. If is the vecto fom Q to any point A on L, then m d sin θ F sinθ dˆ F Q d sin θ m is called the moment vecto o vecto moment of F about Q 7

28 Moe Applications Toque T x F Foce on a moving chage due to a magnetic field due to F q v x B F q B v Velocity of a otating body v w x ω angula speed w ω and diected along axis of otation o w ω v 8

29 9 Two non-zeo vectos A and B ae paallel if & if : 0 B A j k i i j k k i j j i k i k j k j i ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Cicula Mnemonic: Moe Coss Poduct Popeties: Coss poduct is not commutative: A B B A A B B A Coss poduct is not associative: C B) (A C) (B A Example (p7.4, # 3): A <,7,-4>, B <,,-> Find a vecto that is pependicula to A and B

30 7.5 Lines and Planes Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: 4. Section 7.5 expess a line as a: vecto paameteization, and scala paameteization, use vectos to detemine whethe two lines intesect, and if so, the point of intesection. use vectos to find the distance fom a point to a line. expess a plane as a scala equation and as a vecto equation. find whethe two planes intesect, and if so, the angle of intesection and a vecto paameteization of the line fomed by the intesection. unit nomal fo a plane. 30

31 3 Equation of a staight Line > < > <,, : P &,, : P z y x z y x a ta t t the line is in the diection of, scala paamete ), ( + + z z z z y y y y x x x x. Vecto equation of the line though & :. Paametic & symmetic equations of the line: Given two points in 3D: Two foms fo the line though P & P : If a is a unit vecto, then its components ae diection cosines of the line. Examples to follow:

32 Examples: 7.5, # 3 Find the vecto equation of a line though: (/, -/, ) & (-3/, 5/, -/). + t ), t ( scala paamete Examples: 7.5, # 7 Show that the two lines: t <,,> and <6,6,6> + t <-3,-3,-3> ae the same. 3

33 Two foms: 7.5 Equation of a Plane. The equation of a plane pependicula to a nomal vecto is given by: n a iˆ + b ˆj + ( - ) n 0 c kˆ a x + b y + cz + d 0. The equation of a plane contains 3 points: P ( ), P ( ), P 3 ( 3 ) is given by: [ ) ( )] ( ) 0 ( 3 a vecto fom. Examples to follow: 33

34 Examples: 7.5, # 39 Find the equation of a plane contains: (5,,3) & pependicula to <,-3,4> Two methods: a x + b y + cz + d 0 O: < - >. n 0 Answe: Examples: 7.5, # 5 Find the equation of a plane contains: (,3,-5) & paallel to x + y - 4z 34

35 Intesection of Two Planes Let a x + b y + c z d & a x + b y + c z d be two non paallel planes. We get a system of two equations and thee unknowns. Choose one vaiable abitay, say x t, and solve the new system of two equations and two unknowns y and z. paametic equations fo the line of intesection 35

36 Example Find the paametic equation fo the line of intesection of x 3y + 4z and x y z 5 Solution Let choose z t, sub in the equatins and solve fo x and y fom x 3y 4t and x y 5 + t Then, x 4 + 7t, y 9 + 6t, z t END of selected mateials fom Chapte 7. 36