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[Virtual Presenter] Chapter: 3 Vectors. Chapter: 3 Vectors.

[Audio] Coordinate Systems Many aspects of physics involve a description of a location in space. In two dimensions, this description is accomplished with the use of the Cartesian coordinate system, in which perpendicular axes intersect at a point defined as the origin. Cartesian coordinates are also called rectangular coordinates..

[Audio] In this polar coordinate system, r is the distance from the origin to the point having Cartesian coordinates (x, y) and theta is the angle between a fixed axis and a line drawn from the origin to the point. The fixed axis is often the positive x axis, and theta is usually measured counterclockwise from it..

[Audio] Therefore, starting with the plane polar coordinates of any point, we can obtain the Cartesian coordinates by using the equations:.

Activity:.

Use Equation 3.4 to find r. Use Equation 3.3 to find 0: 4.30 m Y -2.50m = 0514 tan0=—= x —3.50 m O = 216 Notice that you must use the signs of x and y to find that the point lies in the third quadrant of the coordinate sy tem. That is, 0 = 2160, not 35.50..

[Audio] VectorandScalarQuantities Ascalarquantityiscompletelyspecifiedbyasinglevaluewithanappropriateunitandhasnodirection Avectorquantityiscompletelyspecifiedbyanumberandappropriateunitsplusadirection. SomePropertiesofVectorsEqualityofTwoVectors(iftheyhavethesamemagnitudeandiftheypointinthesamedirection.Thatis,onlyifABandifandpointinthesamedirectionalongparallellines) AddingVectors(addingvectorsareconvenientlydescribedbyagraphicalmethod).

ACTIVE FIGURE 3.6 IAhen vector Ii is added to vector Å, the resultant is the vector that runs from the tail of Ä to the tip of g. Sign in at www.thomsonedu.com and go to ThomsonNOW to explore the addition of two vectors. c Figure 3.7 Geometric construction for summing four vectors. The resul- tant vector is by definition the one that completes the polygon. Figure 3.8 This construction shows that Ä + = É + A or, in other words, that vector addition is commutative..

resultant displacement. Use R2 = A2 + B2 — 2AB cos 9 from the law of cosines to find R: Substitute numerical values, noting that 0 = 1800 — 600 = 1200: Use the law of sines (Appendix B.4) to find the direction of measured from the northerly direction: EXAMPLE 3.2 A car travels 20.0 km due north and then 35.0 km in a direction 60.00 west of north as shown in Figure 3.1 la. Find the magnitude and direction of the car's sin 1200 = 0.629 A Vacation Trip y (km) w 60.00 s x (km) —20 (a) y (km) 20 x (km) —20 (b) AZ + 132 — 2AB cos 0 (20.0 km)2 + (35.0 km)2 — 2(20.0 km)(35.0 km) cos 1200 48.2 km sin sin O B R 35.0 km sin ß = — sin O = 48.2 km ß 38.90 Figure 3.11 (Example 3.2) (a) Graphical method for finding thel resul- tant displacement vector = Ä + G. (b) Adding the vectors in reverse order (É + A) gives the same result for R..

[Audio] ComponentsofaVectorandUnitVectorsThegraphicalmethodofaddingvectorsisnotrecommendedwheneverhighaccuracyisrequiredorinthree-dimensionalproblems.Inthissection,wedescribeamethodofaddingvectorsthatmakesuseoftheprojectionsofvectorsalongcoordinateaxes.Theseprojectionsarecalledthecomponentsofthevectororitsrectangularcomponents.Anyvectorcanbecompletelydescribedbyitscomponents..

Consider a vector A lying in the xy plane and making an arbitrary angle 0 with the positive x axis as shown in Figure 3.12a. This vector can be expressed as the sum of two other component vectors Ax, which is parallel to the x axis, and Äy, which is parallel to the y axis. From Figure 3.12b, we see that the three vectors form a right triangle and that A = Ax + Ay. We shall often refer to the "components of a vector A," written Ax and A (without the boldface notation). The component Ax represents the projection of A along the x axis, and the component A represents the projection of A along the y axis. These components can be positive or nega- tive. The component Ax is positive if the component vector Ax points in the posi- tive x direction and is negative if Ax points in the negative x direction. The same is true for the component Ay. From Figure 3.12 and the definition of sine and cosine, we see that cos 0 Ax/ A and that sin 0 = AJA. Hence, the components of A are Ax A cos 0 Ay = A sin 0 (3.8) (3.9).

From Figure 3.12 and the definition of sine and cosine, we see that cos 0 = Ax/ A and that sin 0 = AN/ A. Hence, the components of A are (3.8) (3.9) (a) Ax = A cos 0 Ay = A sin 0 (b) Figure 3.12 (a) A vector lying in the xy plane can be represented by its component vectors Äxand (b) The y component vector Ay can be moved to the right so that it adds to The vector sum of the component vectors is A. These three vectors form a right triangle..

The magnitudes of these components are the lengths of the two sides of a right tri- angle with a hypotenuse of length A. Therefore, the magnitude and direction of A are related to its components through the expressions -1 0 = tan (3.10) (3.11) Notice that the signs of the components Ax and A, depend on the angle 0. For example, if 0 1200, Ax is negative and Ay is positive. If 0 2250, both Ax and A are negative. Figure 3.13 summarizes the signs of the components when Ä lies in the various quadrants. When solving problems, you can specify a vector A either with its components Ax and A or with its magnitude and direction A and 9. Suppose you are working a physics problem that requires resolving a vector into its components. In many applications, it is convenient to express the components in a coordinate system having axes that are not horizontal and vertical but that are still perpendicular to each other. For example, we will consider the motion of objects sliding down inclined planes. For these examples, it is often convenient to orient the x axis parallel to the plane and the y axis perpendicular to the plane. Ax negative Ax positive Ay positive Ay positive x Ax negative Ax positive Ay negative Ay negative Figure 3.13 The signs of the compo- nents of a vector Ä depend on the quadrant in which the vector is located..

[Audio] UnitVectors. Unit Vectors.

or (3.14) Because = Rx; + Ryj , we see that the components of the resultant vector are Rx=Ax+Bx (3.15) The magnitude of and the angle it makes with the x axis from its components are obtained using the relationships tan 0 = (3.16) (3.17) We can check this addition by components with a geometric construction as shown in Figure 3.16. Remember to note the signs of the components when using either the algebraic or the graphical method. At times, we need to consider situations involving motion in three component directions. The extension of our methods to three-dimensional vectors is straight- forward. If A and both have x, y, and z components, they can be expressed in the x form The sum of A and is (3.18) (3.19) (3.20) Fig u re 3.16 This geomnetx-ic con- struction for the surn of two vectors shows the velationship between the cornponen ts of the resultant and the cornponents of the individual vec toys. Notice that Equation 3.20 differs from Equation 3.14: in Equation 3.20, the resultant vector also has a z component R: = Az + Bz. If a vector has x, y, and z compo- nents, the magnitude of the vector is R = Rx2 + R, 2 + RZ 2. The angle Ox that makes with the x axis is found from the expression cos Ox = Rx/ R, with similar expressions for the angles with respect to the y and z axes..

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