27 ft 3 adequately describes the volume of a cube with side 3. ft F adequately describes the temperature of a person.

Transcription

1 VECTORS The stud of ectors is closel related to the stud of such phsical properties as force, motion, elocit, and other related topics. Vectors allow us to model certain characteristics of these phenomena with numbers that tell us their magnitude and direction. SCALAR QUATITIES: Measurements inoling such things as time, area, olume, energ, and temperature are called scalar measurements because each can be described adequatel using their magnitude alone (with the appropriate units). 7 ft 3 adequatel describes the olume of a cube with side 3 ft F adequatel describes the temperature of a person. 7ft 3 and 98 0 F are called scalars. Some properties such as force, elocit, and displacement require both magnitude and direction to be described completel. These quantities are called ector quantities. Eample: You are driing due north at 45 miles per hour. The magnitude is the speed, 45 miles per hour. The direction of motion is due north. A C NOTATION AND GEOMETRY OF VECTORS Two airplanes trael at 400 mph on a parallel course and in the same direction. This situation can be modeled using directed line segments. A B D The directed segments are drawn parallel with arrowheads pointing the same wa to indicate direction of flight, while making them the same length indicates that the elocities are the same. The length of the ector models the magnitude of the elocit, while the arrowhead indicates the direction of trael. Vectors are named using the initial and terminal points that define them as in AB and CD Or with a bold, small case letter such as u or. We ma also write them as uor.

2 Q P The magnitude of the directed line segment PQ is its length. We indicate this b PQ. PQ is the distance from point P to point Q. Reference Angle θ r : For an angle θ in standard position, the acute angle θ r formed b the terminal side and the -ais is called the reference angle for θ. Find the reference angle, θ r, for each of the following angles. a) θ = b) θ = c) a) θ r = 5 0 b) θ r = 45 0 c) θ r = 40 0 Eample ( 4, 4) (, ) 3 4 u (,3) Show that u=. Vectors that are equal hae the same magnitude and direction. Use the distance formula to show that u and hae the same magnitude. d = ( ) + ( ) u = [0 ( 3)] + [3 ( 3)] = = (3 0) + (6 0) Thus u and hae the same magnitude: u = = 3 3

3 One wa to show that uand hae the same direction is to find the slopes of the lines on which the lie. m = Verif to show that each ector has a slope of 3/. POSITION VECTORS For a ector with initial point (, ) and terminal point (, ), the position ector for is -,- =, an equialent ector with initial point (0,0) and terminal point (, ). ector in component form is denoted The as ab,, where a is the horizontal component and b is the ertical component. (-, 3) u 7 3 (-5, -4) (3, 7) Find the position ector for ector u and graph it. The position ector for u is: = - - (-5), 3 - (-4) = 3, 7 The position ector has (0,0) as its initial point, and (3, 7) as its terminal point. For a position ector = a, b shown below at left and angle θ r, obsere the following: The magnitude of ector = ertical component: b sinθ r = horizontal component: cosθ r = tanθ r b a b = a a + b b = sinθ r a = θr cosθ r b = tan a θ r a 3

4 Finding the Magnitude and Direction Angle of a Vector Eample: For =.5, 6 and 3 3, 3, a. Find their magnitudes. 6 c. For : θr = tan.5 = tan (. 4) d. For : θr = tan = tan b. Graph each ector and name the quadrant where located. = 3 3,3 in QI. c. Find the angle θ for each ector (round to tenths of a degree). a. = = = 4.5 = 65. ( ) ( ) , 3 =.5, 6 is located in QIII. Wh b. = = = 36 = 6 ( ).5, 6 Find the Horizontal and Vertical Components of a Vector. The ector = ab, is in QIII, has a magnitude of =, and forms 0 an angle of 5 with the negatie -ais. Graph the ector and find its horizontal and ertical components Note: θ r = 5⁰, therefore θ = = Horizontal Component: a θ = 0 = cos cos05 9 Vector Addition: Addition of ectors using the tail to tip method. Shift one ector (without changing its direction) so that its tail (initial point) is at the tip (terminal point) of the other ector. Gien ectors u and Tail of to tip of u Tail of u to tip of u u u + u u + = -8.9 Vertical Component: b = θ = 0 sin b sin u 4

5 Vector Addition: Add ectors and w. Initial point of +ww w Terminal point of w APPLICATION OF VECTORS: A common eample of a ector quantit is force. Other ector quantities that appear in engineering mechanics are moment, displacement, elocit, and acceleration. FORCE: The effect of one phsical bod on another phsical bod. The force effect between two bodies can be interpreted as a push or pull of one of the bodies on the other bod. RESULTANT FORCE: When two or more forces are added to obtain a single force, it produces the same effect as the original sstem of forces. This single ector is called the sum, or the resultant force of the original sstem of forces. The resultant force, F R, is the sum of F and df. F R COMPONENTS OF A FORCE: Two or more forces acting on a particle ma be replaced b a single force which has the same effect on the particle. Conersel, a single force F acting on a particle ma be replaced b two or more forces which, together, hae the F same effect on the particle. These forces are called the F components of the original force F. 5

6 =t-/4/0 RECTANGULAR COMPONENTS OF A FORCE. Often it is desirable to resole a force into two components which are perpendicular to each other. In the figure below, the force F has been resoled into a component F and a component F. F and F are called rectangular components. Eample: A force of 800 N is eerted on a bolt A as shown. Determine the horizontal and ertical components of the force. Note: θ R = 35 0 ; What is θ? F = 800 N F 0 θ F F F = Fcosθ F = FtanθFsinθ=FFθ F an 0 F = 800 N F θ = 35 A F A F = Fcos45 0 = 800cos45 0 = -655 N F = Fsin45 0 = 800sin450 = 459 N We write F in the form F = -(655N)i + (459 N) j Vectors in the Rectangular Coordinate Sstem 6

7 The i and j Unit Vectors. b P = (a, b) Vector i is the unit ector (ector of length ) whose direction is along the positie ais. Vector j is the unit ector whose direction is along the positie ais. 0 = ai + bj a j Vectors in the rectangular coordinate sstem can be represented in terms of i and j. i A unit ector is defined to be a ector whose magnitude is one. In man applications, it is useful to find the unit ector that has the same direction as a gien ector. For an nonzero ector, the ector is the unit ector that has the same direction as. To find this ector, diide b its magnitude. Eample: Find the unit ector in the same direction as = 5i j. Then erif that the ector has magnitude. = 5 + ( ) = = 69 = 3 5 i j 5 = = u j Verif : = + = = = ( ) ( ) ( ) ( ) 7

8 Sketch the ector and find its magnitude. = 3i + 4j a = 3 and b = 4. ( 3, 4) = 5 Eample Sketch the ector =-i+j and find its magnitude

9 Eample 4 Let be a ector from initial point P (, ) to terminal pont P (,3). Write in terms of i and j. Operations with Vectors in Terms of i and j Vector Subtraction: The difference of two ectors, u is defined as u = u + (-): The terminal point of u coincides with the initial point of - u u 9

10 Eample If =i-3j and w=-i+4j find each of the following: a. +w b. -w c. d. -3w e. -3w Unit Vectors 0

11 Eample Find the unit ector in the same direction as =-3i-4j. Then erif that the ector has the magnitude of. Writing a Vector in Terms of Its Magnitude and Direction Eample 0 The wind is blowing at 6 mph in the direction of N60 E. Epress its elocit as a ector in terms of i and j.

12 Application Eample Two forces F and F, of magnitude 5 lbs and lbs, respectiel act on an object. The direction of F is N0 E and the direction of F is N75 E. Find the 0 0 magnitude and the direction of the resultant force. Epress the magnitude to the nearest hundredth of a pound and the direction angle to the nearest tenth of a degree. If u=- 3i+4j, =i-j find u+ (a) (b) -i+3j 4i-j (c) 5i+3j (d) -4i+7j

13 dianse/4/0 Conerting from Rectangular Coordinates to Polar Coordinates. Find the magnitude of the ector =-3i+4j (a) 5 (b) 6 (c) 7 (d) 8 a) Radians. b) Polar Coordinates. A) RADIAN MEASURE We measure angles b determining the amount of rotation from the initial side to the terminal side. Two units of measurement for angles are degrees and radians. Radian Measure An angle whose erte is at the center of a circle is called a central angle. A central angle intercepts the arc of the circle from the initial side to the terminal side. A positie central angle that intercepts an arc of the circle of length equal to the radius of the circle has a measure of radian radian. r ө r RELATIONSHIP BETWEEN DEGREES AND RADIANS: 360 = radians 80 = radians θ=θdegrees to radians: ra 80 How man radians are there in a ocircle? θradians=θ dehow man degrees are there in to degrees: Radians radian? o grs3

14 4 Conert each angle from degrees to radians. a) 30 b) 90 c) -5 d) = = i ra)ad3030ians6= radb)90ians= 5-rc)-ad5ians455 d) = = 0.96radians i Conert each angle in radians to degrees. 3 a) radians b) = - radians c).5radians a) radians 3 3 = = i o b) - radians = = i o-3580 c) radians = = i o4.6the foundation of the polar coordinate sstem is a horizontal ra that etends to the right. This ra is called the polar ais. The endpoint of the polar ais is called the pole. A point P in the polar coordinate sstem is designated b an ordered pair of numbers (r, θ). i th di t d di t f th r pole polar ais θ P = (r, θ) r is the directed distance form the pole to point P ( positie, negatie, or zero). θ is angle from the pole to P (in degrees or radians). 00ToplotthepointP(r,θ),goadistanceofrat0thenmoeθalongacircleofradiusrPlotting Points in Polar Coordinates.0.Ifr>0,plotapointatthatlocation.Ifr<0,thepointisplotedonacircleofthesameradius,but80intheoppositedirection.

15 3,-,-3/4/0 C Plot each point (r, θ) a) A(3, 45 0 ) A b) B(-5, 35 0 ) CONVERTING BETWEEN POLAR AND RECTANGULAR FORMS CONVERTING FROM POLAR TO RECTANGULAR COORDINATES. To conert the polar coordinates (r, θ) of a point to rectangular coordinates (, ), use the equations = rcosθ and = rsinθ B c) C(-3, -/6) Conert the polar coordinates of each point to its rectangular coordinates. a) (, -30 ⁰ ) b) ( 4, /3) 3 a) = rcos(-30⁰) = ( ) = 3 = sin( 30 ) = ( / ) = The rectangular coordinates of (, = -4 sin(/3) = 3 4( ) = 3 30 ) are (The rectangular coordinates of (-4, ) are (3 )b) = -4cos(/3) = -4(/) = - -POLAR COORDINATES: To conert the rectangular coordinates (, ) of a point to polar coordinates: ) Find the quadrant in which the gien point (, ) lies. ) Use r = + to find r. 3) Find θ b using tanθ = and choose θ so that it lies in the same quadrant as the point (, ). )CONVERTING FROM RECTANGULAR TO 5

16 Find the polar coordinates (r, θ) of the point P with r > 0 and 0 θ, whose rectangular coordinates are (, ) = (, 3) The point is in quadrant. r = + = = ( ) ( 3) 4 3 tanθ = θ tan 3 60 r = = θ = = 0 or 0 ( ) = 80 3 The required polar coordinates are (, /3) 6