Chapter 3 Vectors. m = m1 + m2 = 3 kg + 4 kg = 7 kg (3.1)


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1 COROLLARY I. A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately. Isaac Newton  Principia 3.1 Introduction Of all the varied quantities that are observed in nature, some have the characteristics of scalar quantities while others have the characteristics of vector quantities. A scalar quantity is a quantity that can be completely described by a magnitude, that is, by a number and a unit. Some examples of scalar quantities are mass, length, time, density, and temperature. The characteristic of scalar quantities is that they add up like ordinary numbers. That is, if we have a mass m1 = 3 kg and another mass m2 = 4 kg then the sum of the two masses is m = m1 + m2 = 3 kg + 4 kg = 7 kg (3.1) A vector quantity, on the other hand, is a quantity that needs both a magnitude and a direction to completely describe it. Some examples of vector quantities are force, displacement, velocity, and acceleration. The velocity of a car moving at 50 km per hour (km/hr) due east can be represented by a vector. Velocity is a vector because it has a magnitude, 50 km, and a direction, due east. A vector is represented in this text book by boldface script, that is, A. Because we cannot write in boldface script on notepaper or a blackboard, a vector is written there as the letter with an arrow over it. A vector can be represented on a diagram by an arrow. A picture of this vector can be obtained by drawing an arrow from the origin of a Cartesian coordinate system. The length of the arrow represents the magnitude of the vector, while the direction of the arrow represents the direction of the vector. The direction is specified by the angle θ that the vector makes with an axis, usually the xaxis, and is shown in figure 3.1. y A θ Figure 3.1 Representation of a vector. x
2 The magnitude of vector A is written as the absolute value of A namely A, or simply by the letter A without boldfacing. One of the defining characteristics of vector quantities is that they must be added in a way that takes their direction into account. 3.2 The Displacement Probably the simplest vector that can be discussed is the displacement vector. Whenever a body moves from one position to another it undergoes a displacement. The displacement can be represented as a vector that describes how far and in what direction the body has been displaced from its original position. The tail of the displacement vector is located at the position where the displacement started, and its tip is located at the position at which the displacement ended. For example, if you walk 3 km due east, this walk can be represented as a vector that is 3 units long and points due east. It is shown as d 1 in figure 3.2. This is an example of a displacement vector. Suppose you now walk 4 km due north. This distance of 4 km in a northerly direction can be represented as another displacement vector d 2, which is also shown in figure 3.2. The result of these two displacements is a final displacement vector d that shows the total displacement from the original position. Figure 3.2 The displacement vector. We now ask how far did you walk? Well, you walked 3 km east and 4 km north and hence you have walked a total distance of 7 km But how far are you from where you started? Certainly not 7 km, as we can easily see using a little high school geometry. In fact the final displacement d is a vector of magnitude d and that distance d can be immediately determined by simple geometry. Applying the Pythagorean theorem to the right triangle of figure 3.2 we get and thus, d = d d 2 2 d = (3 km) 2 + (4 km) 2 = 25 km d = 5 km (3.2) 32
3 Even though you have walked a total distance of 7 km, you are only 5 km away from where you started. Hence, when these vector displacements are added d = d 1 + d2 (3.3) we do not get 7 km for the magnitude of the final displacement, but 5 km instead. The displacement is thus a change in the position of a body from its initial position to its final position. Its magnitude is the distance between the initial position and the final position of the body. It should now be obvious that vectors do not add like ordinary scalar numbers. In fact, all the rules of algebra and arithmetic that you were taught in school are the rules of scalar algebra and scalar arithmetic, although the word scalar was probably never used at that time. To solve physical problems associated with vectors it is necessary to deal with vector algebra. 3.3 Vector AlgebraThe Addition of Vectors Let us now add any two arbitrary vectors a and b. The result of adding the two vectors a and b forms a new resultant vector R, which is the sum of a and b. This can be shown graphically by laying off the first vector a in the horizontal direction and then placing the tail of the second vector b at the tip of vector a, as shown in figure 3.3. Figure 3.3 The addition of vectors. The resultant vector R is drawn from the origin of the first vector to the tip of the last vector. The resultant vector is written mathematically as R = a + b (3.4) Remember that in this sum we do not mean scalar addition. The resultant vector is the vector sum of the individual vectors a and b. Although a vector is a quantity that has both magnitude and direction, it does not have a position. Consequently a vector may be moved parallel to itself without changing the characteristics of the vector. Because the magnitude of the moved vector is still the same, and its direction is still the same, the vector is the same. Hence, when adding vectors a and b, we can move vector a parallel to itself until the tip of a touches the tip of b. Similarly, we can move vector b parallel to 33
4 itself until the tip of b touches the tail end of the top vector a. In moving the vectors parallel to themselves we have formed a parallelogram, as shown in figure 3.4. Figure 3.4 The addition of vectors by the parallelogram method. From what was said before about the resultant of a and b, we can see that the resultant of the two vectors is the main diagonal of the parallelogram formed by the vectors a and b, hence we call this process the parallelogram method of vector addition. It is sometimes stated as part of the definition of a vector, that vectors obey the parallelogram law of addition. Note from the diagram that R = a + b = b + a (3.5) that is, vectors can be added in any order. Mathematicians would say vector addition is commutative. 3.4 Vector Subtraction  The Negative of a Vector If we are given a vector a, as shown in figure 3.5, then the vector minus a ( a) is a vector of the same magnitude as a but in the opposite direction. That is, if vector a Figure 3.5 The negative of a vector. points to the right, then the vector a points to the left. Vector a is called the negative of the vector a. By defining the negative of a vector in this way, we can now determine the process of vector subtraction. The subtraction of vector b from vector a is defined as a b = a + ( b) (3.6) In other words, the subtraction of b from a is equivalent to adding vector a and the negative vector ( b). This is shown graphically in figure 3.6 (a). If we complete the parallelogram for the addition of a + b, we see that we can move the vector a b parallel to itself until it becomes the minor diagonal of the parallelogram, figure 3.6(b). 34
5 b a  b a a + b b. a (a) (b) Figure 3.6 The subtraction of vectors. 3.5 Addition of Vectors by the Polygon Method To find the sum of any number of vectors graphically, we use the polygon method. In the polygon method, we add each vector to the preceding vector by placing the tail of one vector to the head of the previous vector, as shown in figure 3.7. The resultant vector R is the sum of all these vectors. That is, R = a + b + c + d (3.7) We find R by drawing the vector from the origin of the coordinate system to the tip of the final vector, as shown in figure 3.7. Although this set of vectors could Figure 3.7 Addition of vectors by the polygon method. represent forces, velocities, and the like, it is sometimes easier for the beginning student to visualize them as though they were displacement vectors. It is easy to see from the figure that if a, b, c, and d were individual displacements, R would certainly be the resultant displacement of all the individual displacements. Vectors are usually added analytically or mathematically. In order to do that, we need to define the components of a vector. 35
6 3.6 Resolution of a Vector into Its Components An arbitrary vector a is drawn onto an x,ycoordinate system, as in figure 3.8. Vector a makes an angle θ with the xaxis. To find the xcomponent ax of vector a, we project vector a down onto the xaxis, that is, we drop a perpendicular from the tip of a to the xaxis. One way of visualizing this concept of a component of a vector is to place a light beam above vector a and parallel to the yaxis. The light hitting vector a will not make it to the xaxis, and will therefore leave a shadow on the xaxis. We call this shadow on the xaxis the xcomponent of vector a and denote it by ax. The component is shown as the light red line on the xaxis in figure 3.8. In the same way, we can determine the ycomponent of vector a, ay, by projecting a onto the yaxis in figure 3.8. That is, we drop a perpendicular from the Figure 3.8 Defining the components of a vector. tip of a onto the yaxis. Again, we can visualize this by projecting light, which is parallel to the xaxis, onto vector a. The shadow of vector a on the yaxis is the y component ay, shown in figure 3.8 as the light red line on the yaxis. The components of the vector are found mathematically by noting that the vector and its components constitute a triangle, as seen in figure 3.9. From trigonometry, we find the xcomponent of a from cos θ = ax a (3.8) 36
7 Figure 3.9 Finding the components of a vector mathematically. Solving for ax, the xcomponent of vector a obtained is ax = a cos θ (3.9) We find the ycomponent of vector a from sin θ = ay a Hence, the ycomponent of vector a is (3.10) ay = a sin θ (3.11) Example 3.1 Finding the components of a vector. The magnitude of vector a is 10.0 units and the vector makes an angle of with the xaxis. Find the components of a. Solution The xcomponent of vector a, found from equation 3.9, is ax = a cos θ = 10.0 units cos = 8.66 units The ycomponent of a, found from equation 3.11, is ay = a sin θ = 10.0 units sin = 5.00 units To go to this interactive example click on this sentence. 37
8 What do these components of a vector represent physically? If vector a is a displacement, then ax would be the distance that the object is east of its starting point and ay would be the distance north of it. That is, if you walked a distance of 10.0 km in a direction that is north of east, you would be 8.66 km east of where you started from and 5.00 km north of where you started from. If, on the other hand, vector a were a force of 10.0 N applied at an angle of to the xaxis, then the xcomponent ax is equivalent to a force of 8.66 N in the xdirection, while the ycomponent ay is equivalent to a force of 5.00 N in the ydirection. 3.7 Determination of a Vector from Its Components If the components a x and ay of a vector are given, and we want to find the vector a itself, that is, its magnitude a and its direction θ, then the process is the inverse of the technique used in section 3.6. The components ax and ay of vector a are seen in figure If we form the triangle with sides ax and ay, then the hypotenuse of that Figure 3.10 Determining a vector from its components. triangle is the magnitude a of the vector, and is determined by the Pythagorean theorem as a 2 = ax 2 + ay 2 (3.12) Hence, the magnitude of vector a is a = a x 2 + a y 2 (3.13) It is thus very simple to find the magnitude of a vector once its components are known. To find the angle θ that vector a makes with the xaxis the definition of the tangent is used, namely tan θ = ay ax 38
9 We find the angle θ by using the inverse tangent, as θ = tan 1 ay ax (3.14) Example 3.2 Finding a vector from its components. The components of a certain vector are given as ax = 7.55 and ay = Find the magnitude of the vector and the angle θ that it makes with the xaxis. The magnitude of vector a, found from equation 3.13, is The angle θ, found from equation 3.14, is Solution a = a x 2 + a y 2 = (7.55) 2 + (3.25) 2 = = 8.22 θ = tan 1 ay = tan = tan ax 7.55 = Therefore, the magnitude of vector a is 8.22 and the angle θ is The techniques developed here will be very useful later for the addition of any number of vectors. To go to this interactive example click on this sentence. The components of a vector can also be found along axes other than the traditional horizontal and vertical ones. A coordinate system can be orientated any way we choose. For example, suppose a block is placed on an inclined plane that makes an angle θ with the horizontal, as shown in figure Let us find the components of the weight of the block parallel and perpendicular to the inclined plane. We draw in a set of axes that are parallel and perpendicular to the inclined plane, as shown in figure 3.11, with the positive xaxis pointing down the plane and the positive yaxis perpendicular to the plane. To find the components parallel and 39
10 Figure 3.11 Components of the weight parallel and perpendicular to the inclined plane. perpendicular to the plane, we draw the weight of the block as a vector pointed toward the center of the earth. The weight vector is therefore perpendicular to the base of the inclined plane. To find the component of w perpendicular to the plane, we drop a perpendicular line from the tip of vector w onto the negative yaxis. This length w is the perpendicular component of vector w. Similarly, to find the parallel component of w, we drop a perpendicular line from the tip of w onto the positive x axis. This length w is the parallel component of the vector w. The angle between vector w and the perpendicular axis is also the inclined plane angle θ, as shown in the comparison of the two triangles in figure Figure 3.12 Comparison of two triangles. (Figure 3.12 is an enlarged view of the two triangles of figure 3.11). In triangle I, the angles must add up to Thus, while for triangle II θ + α = (3.15) 310
11 β + α = (3.16) From equations 3.15 and 3.16 we see that β = θ (3.17) This is an important relation that we will use every time we use an inclined plane. Example 3.3 Components of the weight perpendicular and parallel to the inclined plane. A 100N block is placed on an inclined plane with an angle θ = , as shown in figure Find the components of the weight of the block parallel and perpendicular to the inclined plane. Solution We find the perpendicular component of w from figure 3.11 as The parallel component is w = w cos θ = 100 N cos = 64.3 N (3.18) w = w sin θ = 100 N sin = 76.6 N (3.19) To go to this interactive example click on this sentence. One of the interesting things about this inclined plane is that the component of the weight parallel to the inclined plane supplies the force responsible for making the block slide down the plane. Similarly, if you park your car on a hill with the gear in neutral and the emergency brake off, the car will roll down the hill. Why? You can now see that it is the component of the weight of the car that is parallel to the hill that essentially pulls the car down the hill. That force is just as real as if a person were pushing the car down the hill. That force, as can be seen from equation 3.19, is a function of the angle θ. If the angle of the plane is reduced to zero, then w = w sin 0 0 = 0 Thus, we can reasonably conclude that when a car is not on a hill (i.e., when θ = 0 0 ) there is no force, due to the weight of the car, to cause the car to move. Also note that the steeper the hill, the greater the angle θ, and hence the greater the component of the force acting to move the car down the hill. 311
12 3.8 Unit Vectors It is convenient in our analysis to introduce the concept of unit vectors. A unit vector is a vector that has a magnitude of one, hence the name, unit. It is used to specify a particular direction. The most common of the unit vectors are the i j k system of vectors shown in figure The unit vector i is a unit vector in the z y k j i x Figure 3.13 The i j k system of unit vectors. xdirection, the unit vector j is a unit vector in the ydirection, and the unit vector k is a unit vector in the zdirection. Also notice from figure 3.13, the coordinate system is a righthanded coordinated system. This means that if you place your right hand along the xaxis with your palm facing the yaxis, and then rotate your hand toward the yaxis, your thumb will face in the zdirection. Let us now consider an arbitrary vector a in two dimensions, as shown in figure The components ax and ay of the y y a a = i a x + ja y a y a y ja y ja y a x (a) x ia x (b) x Figure 3.14 A vector in terms of unit vectors. vector a are shown in figure 3.14a. If the xcomponent, ax, is multiplied by the unit vector i, the vector iax is obtained. The vector iax is a vector in the xdirection with magnitude ax as shown in figure 3.14b. Multiplying the ycomponent of the vector a, ay, by the unit vector j yields the vector jay. The vector jay is a vector in the y direction with magnitude ay and is also shown in figure 3.14b. As can be seen from 312
13 figure 3.14b, adding the vector iax to the vector jay yields the original vector a. This can be written mathematically as a= ia + j a (3.20) x Equation 3.20 is the form of the equation used for writing a vector in terms of its components and the unit vectors. We will use this form of the equation for writing many of the vectors in this course. Besides the i j k unit vectors there are other unit vectors specifying other directions. These will be defined later as the need arises. y Example 3.4 Finding the magnitude and direction of a vector when expressed in terms of unit vectors. Given the vector a = 3i + 6j Find the magnitude of the vector a and the direction it makes with the xaxis. Solution The magnitude of the vector is found from equation 3.13 as a = a x 2 + a y 2 = = 45 = 6.71 The direction is found from equation 3.14 as a 6 θ = = = 1 y 1 1 tan tan tan 2 ax 3 θ = To go to this interactive example click on this sentence. 3.9 The Addition of Vectors by the Component Method A very important technique for the addition of vectors is the addition of vectors by the component method. Let us assume that we are given two vectors, a and b, and we want to find their vector sum. The sum of the vectors is the resultant vector R given by R = a + b (3.21) 313
14 and is shown in figure We determine R as follows. First, we find the components ax and ay of vector a by making the projections onto the x and yaxes, respectively. To find the components of the vector b, we again make a projection onto the x and yaxes, but note that the tail of vector b is not at the origin of coordinates, but rather at the tip of a. So both the tip and the tail of b are projected onto the xaxis, as shown, to get bx, the xcomponent of b. In the same way, we project b onto the yaxis to get by, the ycomponent of b. All these components are shown in figure Figure 3.15 The addition of vectors by the component method. The resultant vector R is given by equation 3.21, and because R is a vector it has components R x and Ry, which are the projections of R onto the x and yaxes, respectively. They are shown in figure Now let us go back to the original diagram, figure 3.15, and project R onto the xaxis. Here Rx is shown a little distance below the xaxis, so as not to confuse Rx with the other components that are already there. Similarly, R is projected onto the yaxis to get Ry. Again Ry is slightly displaced from the yaxis, so as not to confuse Ry with the other components already there. Look very carefully at figure Note that the length of Rx is equal to the length of ax plus the length of bx. Because components are numbers and hence add like ordinary numbers, this addition can be written simply as R x = ax + bx (3.22) That is, the xcomponent of the resultant vector is equal to the sum of the x components of the individual vectors. In the same manner, look at the geometry on the yaxis of figure The length Ry is equal to the sum of the lengths of ay and by, and therefore R y = ay + by (3.23) Thus, the ycomponent of the resultant vector is equal to the sum of the ycomponents of the individual vectors. 314
15 The same results can be obtained mathematically by writing the vectors in terms of the unit vectors. That is, a = iax + jay (3.24) and b = ibx + jby (3.25) The addition of the two vectors is simply a + b = iax + jay + ibx + jby gathering the terms with the same unit vectors together we obtain and simplifying a + b = iax + ibx + jay + jby a + b = i(ax + bx) + j(ay + by) (3.26) But the resultant vector R can also be written in terms of the unit vectors as However, since R = a + b, it follows that R = irx + jry (3.27) and therefore, irx = i(ax + bx) Rx = ax + bx (3.28) Notice that equation 3.28, obtained mathematically, agrees with equation 3.22, which was obtained geometrically. The ycomponent of the resultant vector is obtained similarly, i.e., jry = j(ay + by) Ry = ay + by (3.29) Again, notice that equation 3.29, obtained mathematically, agrees with equation 3.23, which was obtained geometrically. We could have performed this entire derivation mathematically, but the geometric approach gives us a more physical picture of what is happening. This demonstration of the addition of vectors was done for two vectors because it is easier to see the results in figure 3.15 for two vectors than it is for many vectors. However, the technique would be the same for the addition of any number of vectors. For the general case where there are many vectors, equations 3.28 and 3.29 for R x and Ry can be generalized to and R x = ax + bx + cx + dx + (3.30) 315
16 Ry = ay + by + cy + dy + (3.31) The plus sign and the dots that appear in equations 3.30 and 3.31 indicate additional components can be added for any additional vectors. We now have Rx and Ry, the components of the resulting vector R, as shown in figure But if the components of R are known the resultant vector can be Figure 3.16 The resultant vector. written as R=iR x +jr y (3.32) If it is desired to determine the magnitude of the resultant vector, R, this can be done by using the Pythagorean Theorem, i.e., R = R x 2 + R y 2 (3.33) The angle θ in figure 3.16 is found from the geometry to be Thus, R tanθ = R θ y x R 1 y = tan (3.34) R x where Rx and Ry are given by equations 3.30 and The magnitude R and the direction θ, of the resultant vector R, have thus been found. Therefore, the sum of any number of vectors can be determined by the component method of vector addition. Example 3.5 The addition of vectors by the component method. Find the resultant of the following four vectors: 316
17 A = 100, θ1 = B = 200, θ 2 = C = 75.0, θ 3 = D = 150, θ 4 = Solution The four vectors are drawn in figure Because any vector can be moved parallel to itself, all the vectors have been moved so that they are drawn as emanating from Figure 3.17 Addition of four vectors. the origin. Before actually solving the problem, let us first outline the solution. To find the resultant of these four vectors, we must first find the individual components of each vector, then we find the x and ycomponents of the resulting vector from R x = Ax + Bx + Cx + Dx (3.35) Ry = Ay + By + Cy + Dy (3.36) We then find the resulting vector from and R = R x 2 + R y 2 θ = tan 1 Ry Rx (3.37) (3.38) The actual solution of the problem is found as follows: we find the individual x components as Ax = A cos θ1 = 100 cos = 100(0.866) = 86.6 B x = B cos θ2 = 200 cos = 200(0.500) = C x = C cos θ3 = 75 cos = 75( 0.766) =
18 Dx = D cos θ4 = 150 cos = 150( 0.342) = 51.3 R x = Ax + Bx + Cx + Dx = 77.8 whereas the ycomponents are Ay = A sin θ1 = 100 sin = 100(0.500) = 50.0 B y = B sin θ2 = 200 sin = 200(0.866) = C y = C sin θ3 = 75 sin = 75(0.643) = 48.2 D y = D sin θ4 = 150 sin = 150( 0.940) = R y = Ay + By + Cy + Dy = The x and ycomponents of vector R are shown in figure Because Rx and Ry are both positive, we find vector R in the first quadrant. If Rx were negative, R would have been in the second quadrant. It is a good idea to plot the components Rx and Ry for any addition so that the direction of R is immediately apparent. Figure 3.18 The resultant vector. The resultant vector can be written in terms of the unit vectors as R = irx + jry = 77.8i + 130j The magnitude of the resultant vector is found from equation 3.37 as R = R 2 x + R 2 y = (77.8) 2 + (130.2) 2 = 23, = 152 The angle θ that vector R makes with the xaxis is found as θ = tan 1 Ry = tan = tan Rx 77.8 = as is seen in figure
19 It is important to note here that the components Cx, Dx, and Dy are negative numbers. This is because Cx and Dx lie along the negative xaxis and Dy lies along the negative yaxis. We should note that in the solution of the components of the vector C in this problem, the angle of was entered directly into the calculator to give the solution for the cosine and sine of that angle. The calculator automatically gives the correct sign for the components if we always measure the angle from the positive xaxis. 1 To go to this interactive example click on this sentence. Example 3.6 Vectors in terms of unit vectors. Given the two vectors a = 3i + 2j b = 6i + 3j Find the following vector, its magnitude and its direction for (a) a + b, (b) a b, and (c) b a. a. The sum of the two vectors is found as Solution Its magnitude is Its direction is found as a + b = (3 + 6)i + (2 + 3)j a + b = 9i + 5j a + b = = 106 = 10.3 R 5 θ = = = 1 y 1 1 tan tan tan Rx 9 0 θ = We can also measure the angle that the vector makes with any axis other than the positive xaxis. For example, instead of using the angle of with respect to the positive xaxis, an angle of 40 0 with respect to the negative xaxis can be used to describe the direction of vector C. The xcomponent of vector C would then be given by Cx = C cos 40 0 = 75.0 cos 40 0 = Note that this is the same numerical value we obtained before, however the answer given by the calculator is now positive. But as we can see in figure 3.17, Cx is a negative quantity because it lies along the negative xaxis. Hence, if you do not use the angle with respect to the positive xaxis, you must add the positive or negative sign that is associated with that component. In most of the problems that will be covered in this text, we will measure the angle from the positive xaxis because of the simplicity of the calculation. However, whenever it is more convenient to measure the angle from any other axis, we will do so. 319
20 b. The difference between the two vectors is Its magnitude is a b = (3 6)i + (2 3) a b = 3i j a b = ( 3) 2 + ( 1) 2 = 10.0 = 3.16 Its direction is found as R 1 θ = = = R 3 1 y 1 1 tan tan tan x θ = Notice that both Rx and Ry are negative indicating that the angle θ is an angle in the third quadrant, between the negative xaxis and the vector (a b). The angle that the vector (a b) makes with the positive xaxis is found by adding to the angle θ, giving an angle of with respect to the positive xaxis. c. The difference between the two vectors is b a = (6 3)i + (3 2)j b a = 3i + j = (a b) Its magnitude is the same as the magnitude of a b, Its direction is found as b a = ( 3) 2 + (1) 2 = 10.0 = 3.16 R 1 θ = = = R 3 1 y 1 1 tan tan tan x θ = Notice that in this case both Rx and Ry are positive indicating that the angle θ is in the first quadrant, and hence the angle should be considered as positive. To go to this interactive example click on this sentence. Although most problems treated in this text will be two dimensional, we do live in a three dimensional world and hence most vectors are 3dimensional. For those problems requiring threedimensional vectors the extension to three dimensions is 320
21 quite simple. A three dimensional vector a is shown in figure Notice that it has the component ax in the xdirection, ay in the ydirection, and az in the z direction. z a a z a x y x a y Figure 3.19 A vector in three dimensions. The vector a is written mathematically as a = iax + jay + kaz (3.39) while another arbitrary vector, b, is written as b = ibx + jby + kbz (3.40) The sum of the two vectors is determined as before R = a + b = iax + jay + kaz + ibx + jby + kbz R = a + b = i(ax + bx ) + j(ay + by) + k(az + bz) (3.41) And the resultant vector R can also be written in terms of the unit vectors as And as determined for the two dimensional case, R = irx + jry + krz (3.42) Rx = ax + bx (3.43) Ry = ay + by (3.44) Rz = az + bz (3.45) Finally, the magnitude of the resultant vector is found by the extension of the Pythagorean theorem to three dimensions as 321
22 R = R x 2 + R y 2 + R z 2 (3.46) Example 3.7 The magnitude of a three dimensional vector. Given the vector Find the magnitude of the vector a. a = 4i + 3j + 6k Solution The magnitude of the vector is found from equation 3.46 as a = a x 2 + a y 2 + a z 2 = a = 61 = 7.81 To go to this interactive example click on this sentence. Example 3.8 The resultant of threedimensional vectors. Given the two vectors a = 7i + 2j + 8k b = 4i + 5j 3k Find the resultant vector and its magnitude. The sum of the two vectors is found as Solution R = a + b = (7 + ( 4))i + (2 + 5)j + (8 3)k R = a + b = 3i + 7j + 5k Its magnitude is found from equation 3.46 as R = R 2 x + R 2 2 y + R z R = = 83.0 = 9.11 To go to this interactive example click on this sentence. 322
23 3.10 The Multiplication of Vectors Not only can vectors be added and subtracted they can also be multiplied. The multiplication of vectors is important because there are many physical laws that are expressed as the product of vectors. There are three types of multiplication involving vectors. They are: 1) The multiplication of a vector by a scalar. Consider the vector a in figure 3.20a. This vector has both magnitude and direction. If a is multiplied by a scalar k then the product ka is a new vector, say b, where b = ka If k is a positive number greater than one, then the product represents a vector in the same direction as a but elongated by a factor k, as shown in figure 3.20b. If k is less than one, but greater than zero, the new vector is shorter than a, figure 3.20c, and if k is negative, the new vector is in the opposite direction of a, figure 3.20d. (a) a (b) ka k > 1 (c) ka 0 < k < 1 (d) ka k < 0 Figure 3.20 Multiplication of a vector by a scalar. 2) The scalar product. The scalar product is the multiplication of two vectors, the result of which is a scalar. The scalar product will be discussed in detail in the next section. 3) The vector product. The vector product is the multiplication of two vectors, the result of which is a vector. The vector product will be discussed in detail in section The Scalar Product or Dot Product The scalar product of two vectors a and b, figure 3.21, is defined to be ab= abcosθ (3.47) 323
24 b θ a Figure 3.21 The scalar product. where θ is the angle between the two vectors when they are drawn tailtotail, and a and b are the magnitudes of the two vectors. The symbol for this type of multiplication is the dot,, between the vectors a and b. Hence, the scalar product is also called the dot product. The definition, equation 3.47 might seem quite arbitrary with the inclusion of the cosθ. You might ask, why not use the tanθ or some other trigonometric function? The answer is quite simple; as will be shown shortly, equation 3.47 can represent different physical concepts and phenomena. As can be observed in figure 3.21, b cosθ is the component of the vector b in the a direction. Thus, the scalar product is the component of the vector b in the direction of the vector a, multiplied by the magnitude of the vector a itself. It can also be stated that the quantity, a cosθ, is the component of the vector a in the b direction and hence the scalar product is also the component of the vector a in the b direction, multiplied by the magnitude of the vector b itself. Either description is correct. The inclusion of the cosθ in the definition of the scalar product gives rise to some interesting special cases. If the vectors a and b are parallel to each other, then the angle θ between a and b is equal to zero. In that case, the scalar product becomes a b = ab cos0 0 But the cos0 0 = 1, therefore a b = ab (for a b) (Note that as a special case, a a = a 2, and hence a = aa defines the magnitude of any vector.) On the other hand, if a is perpendicular to b, then the angle between the two vectors is 90 0, and the scalar product becomes But the cos90 0 = 0, and hence a b = ab cos 90 0 a b = 0 (for a b) As an example, if the vector a has a magnitude of 2 units and vector b a magnitude of 3 units, then their dot product can take on any value between 6 and +6 depending on the angle θ between the two vectors. For scalars, on the other hand, 2 times 3 is always equal to 6 and nothing else. 324
25 Example 3.9 The scalar or dot product. The vector a has a magnitude of 5.00 units and the vector b a magnitude of 7.00 units. If the angle between the vectors is , find their scalar product. Solution The scalar product of the two vectors is found, by equation 3.47, to be a b = ab cosθ a b = (5.00)(7.00)cos a b = 21.1 To go to this interactive example click on this sentence. If the vectors are expressed in terms of the unit vectors, the dot product becomes a b = (iax + jay + kaz) (ibx + jby + kbz) The dot product of the unit vectors are a b = i iaxbx + i jaxby + i kaxbz + j iaybx + j jayby + j kaybz (3.48) + k iazbx + k jazby + k kazbz i i = (1)(1)cos0 0 = 1 j j = (1)(1)cos0 0 = 1 k k = (1)(1)cos0 0 = 1 (3.49) i j = j i = (1)(1)cos90 0 = 0 i k = k i = (1)(1)cos90 0 = 0 k j = j k = (1)(1)cos90 0 = 0 Notice that the dot product of a unit vector by itself is equal to one, because the magnitude of the unit vectors is equal to one and the angle between the unit vector and itself is equal to zero, and of course, the cosine of zero degrees is equal to one. Also notice that the dot product of two different unit vectors is equal to zero, because the angle between two different unit vectors is 90 0 and the cosine of 90 degrees is equal to zero. Using the results of equations 3.49, the dot product of two vectors becomes ab= ab + ab + ab (3.50) x x y y z z A special case of equation 3.50 is the dot product of a vector by itself. That is, 325
26 a a = axax + ayay + azaz aa= a + a + a = a (3.51) x y z and hence the magnitude of any vector is given by, a = a x 2 + a y 2 + a z 2 (3.52) Example 3.10 The dot product of threedimensional vectors. Find the dot product of the following vectors a = 3i + 5j + 2k b = 4i + 7j + 5k Solution The dot product is found from equation 3.50 as a b = axbx + ayby + azbz a b = (3)( 4) + (5)(7) + (2)(5) a b = 33 Notice that the result of the dot product of two vectors is a scalar, that is, a number. To go to this interactive example click on this sentence The Vector Product or Cross Product The magnitude of the vector product of two vectors a and b is defined to be a b =absinθ (3.53) where θ is the angle between the two vectors when they are drawn tailtotail, and is shown in figure The symbol for the multiplication is designated by the 326
27 z a x b b? y a x Figure 3.22 The cross product. cross sign,, between the two vectors, and hence the name, cross product. The result of the cross product of two vectors, a and b, is another vector, a b, which is perpendicular to the plane made by the vectors a and b. The direction of a b is found by taking your right hand with the fingers in the same direction as the vector a, your palm facing toward the vector b, and then rotating your right hand through the angle between a and b. Your thumb will then point in the direction of a b as seen in figure The new vector is both perpendicular to the vector a and the vector b. This rule is called the right hand rule and it is imperative that the right hand be used. a x b b a Figure 3.23 The right hand rule to determine the direction of the cross product. One of the characteristics of this vector product, is that the order of the multiplication is very important, for as can be seen from figure 3.24 θ a b= b a (3.54) a xb b θ a a Figure 3.24 The sequence of the cross multiplication is very important. 327
28 The magnitude of a cross b is the same as the magnitude of b cross a from the defining equation 3.53, but their directions are opposite to each other as seen in figure This result is often stated as: vectors are noncommutative under vector multiplication. If the vectors a and b are parallel to each other then θ = 0 0 and But sin0 0 = 0, therefore a b = ab sin0 0 a b = 0 (for a b) (3.55) If the vector a is perpendicular to the vector b, then θ = 90 0 and But the sin 90 0 = 1, therefore a b = ab sin 90 0 a b = ab (for a b) (3.56) Note that these two cases are the opposite of what was found for the dot product. This is because the dot product contains the cosθ term, while the cross product contains the sinθ. The cosθ and sinθ are complementary functions and they are out of phase with each other by Example 3.11 The vector or cross product. The vector a has a magnitude of 5.00 units and the vector b of 7.00 units. If the angle between the vectors is , find the magnitude of their cross product. Solution The magnitude of their cross product is found from equation 3.53 to be a b = ab sinθ a b = (5.00)(7.00)sin a b = 28.0 The direction of a b is as shown in figure To go to this interactive example click on this sentence. The area of a surface can be represented by the cross product of the two vectors that generate the area. As an example, consider the two vectors, a and b, in figure If vectors a and b are moved parallel to themselves they are still the same vectors, since they still have the same magnitude and direction. But in the process 328
29 A = a x b θ b bsinθ a a b Figure 3.25 The area of a surface. of moving them parallel to themselves they have generated a parallelogram. As you recall from elementary geometry, the area of a parallelogram is equal to the product of its base times its altitude, i.e., A = (base)(altitude) The base of the parallelogram is given by the magnitude of the vector a, and the altitude is given by altitude = b sinθ Therefore, the area of the parallelogram is given by But A = ab sinθ ab sinθ = a b Therefore, the area of a parallelogram generated by sides a and b is A = a b = ab sinθ (3.57) Because a b is a vector perpendicular to the surface generated by the vectors a and b, the area of the surface can be represented as the vector A, where A = a b (3.58) The area vector A is thus perpendicular to the surface generated by the vectors a and b. If the vectors are expressed in terms of the unit vectors, the cross product becomes a b = (iax + jay + kaz) (ibx + jby + kbz) (3.59) a b = i iaxbx + i jaxby + i kaxbz + j iaybx + j jayby + j kaybz + k iazbx + k jazby + k kazbz (3.60) The magnitudes of the cross product of the unit vectors are i i = (1)(1)sin0 0 = 0 j j = (1)(1)sin0 0 =
30 k k = (1)(1)sin0 0 = 0 (3.61) i j = j i = (1)(1)sin90 0 = 1 i k = k i = (1)(1)sin90 0 = 1 k j = j k = (1)(1)sin90 0 = 1 Notice that the cross product of a unit vector by itself is equal to zero, because the magnitude of the unit vector is equal to one and the angle between the unit vector and itself is equal to zero, and of course, the sine of zero degrees is equal to zero. Also notice that the magnitude of the cross product of two different unit vectors is equal to one, because the angle between two different unit vectors is 90 0 and the sine of 90 degrees is equal to one. Because the cross product of two vectors yields another vector, the cross product of two different unit vectors must be another unit vector. It is now necessary to determine the direction of this unit vector. The direction of the new unit vector obtained by the cross product of two unit vectors is obtained with the help of figure 3.26(a). The direction of the cross product of i j is obtained by placing the right hand in the direction of the unit vector i, with the palm facing toward the unit vector j, and then rotating the hand toward the vector j. The thumb will point in the direction of the new vector. But as can be seen in figure 3.26(a), the thumb would point in the direction of the unit vector k. Since the cross product of the unit vectors i and j must be a unit vector, and the direction of this vector has been shown to be in the k direction, the new unit vector must be the unit vector k. Hence, z k = i xj y j = k x i i = j x k x k i j (a) (b) Figure 3.26 Multiplication of the unit vectors. i j = k (3.62) That is, the cross product of the unit vectors i and j is equal to the unit vector k. Using the same technique and figure 3.26(a), it can be seen that and j k = i (3.63) k i = j (3.64) Because the cross product is noncommutative, as shown in equation 3.54, the cross product of j i will point in the negative k direction, as can be seen in figure 3.26(a). Therefore, we also have 330
31 and j i = k (3.65) i k= j (3.66) k j = i (3.67) Using the results of equations 3.61 through 3.67 in equation 3.60 gives the cross product of two vectors as a b = (0)axbx + (k)axby + ( j)axbz + ( k)aybx + (0)ayby + (i)aybz + (j)azbx + ( i)azby + (0)azbz (3.68) Upon gathering like terms together, this becomes a b = i(aybz azby) + j(azbx axbz) + k(axby aybx) (3.69) Every student should make this calculation at least once to see how the cross products of the unit vectors are determined. However, once this has been accomplished it is easy to remember the sequence of the result by using the little circle shown in figure 3.26(b). If you go around the circle in a clockwise manor from i to j to k to i, the cross product of the first two vectors will yield the next unit vector in the cycle. Thus, i j = k; j k = i; and k i = j. If the circle is traversed in a counterclockwise direction, the resultant unit vector will be negative. Hence, i k = j; k j = i; and j i = k. The same cyclic relation can be used to remember the form of equation 3.69 because i is in the xdirection, j is in the y direction, and k is in the z direction. Hence, the cycle i j k is the same as the cycle x y z. The first term in equation 3.69 starts the cycle with i, which is associated with x, the next term in the parenthesis has the subscripts y and z. The minus term within the parenthesis then reverses the subscripts to z and y. The second term in equation 3.69 starts with j, which is associated with y, and the next term in the parenthesis has the subscripts z and x. The minus term within the parenthesis then reverses the subscripts to x and z. Finally the last term in equation 3.69 starts with k which is associated with z, and the next term in the parenthesis has the subscripts x and y. The minus term within the parenthesis then reverses the subscripts to y and x. Hence by remembering the cycle i j k and x y z it is easy to remember the form of equation Example 3.12 The vector product in terms of components. Given the two vectors Find their vector product. a = 3i + 5j + 2k b = 4i + 7j + 5k 331
32 Solution The cross product is found from equation 369 as a b = i(aybz azby) + j(azbx axbz) + k(axby aybx) a b = i{(5)(5) (2)(7)} + j{(2)( 4) (3)(5)} + k{(3)(7) (5)( 4)} a b = 11i 23j + 41k Notice that the result of the cross product is indeed a vector. To go to this interactive example click on this sentence. Example 3.13 The vector product in terms of a determinant. Show that the cross product of two vectors can be represented by a determinant as i j k a b = a a a (3.70) x y z b b b x y z Solution Using the rules for expansion of a determinant, equation 3.70 can be expressed as a a a a a a a b = i b b j b b + k b b y z x z x y y z x z x y a b = i(aybz azby) j(axbz azbx) + k(axby aybx) Factoring a minus one from inside the second term this becomes a b = i(aybz azby) + j(azbx axbz) + k(axby aybx) which is identical to equation Thus, the cross product can also be represented by a determinant. 332
33 The Language of Physics Scalar A scalar quantity is a quantity that can be completely described by a magnitude, that is, by a number and a unit (p. ). Vector A vector quantity is a quantity that needs both a magnitude and direction to completely describe it (p. ). Resultant The vector sum of any number of vectors is called the resultant vector (p. ). Parallelogram method of vector addition The main diagonal of a parallelogram is equal to the magnitude of the sum of the vectors that make up the sides of the parallelogram (p. ). Component of a vector The projection of a vector onto a specified axis. The length of the projection of the vector onto the xaxis is called the xcomponent of the vector. The length of the projection of the vector onto the yaxis is called the ycomponent of the vector (p. ). The addition of vectors by the component method The xcomponent of the resultant vector R x is equal to the sum of the xcomponents of the individual vectors, while the ycomponent of the resultant vector Ry is equal to the sum of the ycomponents of the individual vectors. The magnitude of the resultant vector is then found by the Pythagorean theorem applied to the right triangle with sides Rx and Ry. The direction of the resultant vector is found by trigonometry (p. ). Scalar Product The multiplication of two vectors, the result of which is a scalar (p. ). Vector Product The multiplication of two vectors, the result of which is a vector (p. ). Summary of Important Equations Vector addition is commutative R = a + b = b + a (3.5) Subtraction of vectors a b = a + ( b) (3.6) Addition of vectors R = a + b + c + d (3.7) 333
34 xcomponent of a vector ax = a cos θ (3.9) ycomponent of a vector ay = a sin θ (3.11) Magnitude of a vector Direction of a vector Vector in terms of unit vectors (3.20) a = a 2 2 x + a y θ = tan 1 ay ax a = iax + jay (3.13) (3.14) xcomponent of resultant vector R x = ax + bx + cx + dx... (3.30) ycomponent of resultant vector Ry = ay + by + cy + dy... (3.31) Resultant vector R = irx + jry (3.32) Magnitude of resultant vector R = R x 2 + R y 2 (3.33) Direction of resultant vector θ = tan 1 Ry (3.34) Three dimensional vector a = iax + jay + kaz (3.39) Resultant vector R = irx + jry + krz (3.42) Scalar product a b = ab cosθ (3.47) Scalar product a b = axbx + ayby + azbz (3.50) Rx Magnitude of a vector a = a x 2 + a y 2 + a z 2 (3.52) Magnitude of cross product a b = ab sinθ (3.53) Vector product is noncommutative a b = b a (3.54) Area of a parallelogram A = a b = ab sinθ (3.57) Vector product a b = i(aybz azby) + j(azbx axbz) + k(axby aybx) (3.69) 334
35 i j k Vector product a b = ax ay az (3.70) b b b x y z Questions for Chapter 3 1. Give an example of some quantities that are scalars and vectors other than those listed in section Can a vector ever be zero? What does a zero vector mean? 3. Since time seems to pass from the past to the present and then to the future, can you say that time has a direction and therefore could be represented as a vector quantity? 4. Does the subtraction of two vectors obey the commutative law? 5. What happens if you multiply a vector by a scalar? 6. What happens if you divide a vector by a scalar? 7. If a person walks around a block that is 80.0 m on each side and ends up at the starting point, what is the person s displacement? 8. How can you add three vectors of equal magnitude in a plane such that their resultant is zero? 9. When are two vectors a and b equal? 10. If a coordinate system is rotated, what does this do to the vector? to the components? 11. Why are all the fundamental quantities scalars? 12. A vector equation is equivalent to how many component equations? 13. If the components of a vector a are a x and ay, what are the components of the vector b = 5a? 14. If a + b = a b, what is the angle between a and b? Problems for Chapter Resolution of a Vector into Its Components and Determination of a Vector from Its Components 1. A strong child pulls a sled with a force of 100 N at an angle of 35 0 above the horizontal. Find the vertical and horizontal components of this pull. 2. A 50N force is directed at an angle of 50 0 above the horizontal. Resolve this force into vertical and horizontal components. 3. A girl wants to hold a 68.0N sled at rest on a snowcovered hill. The hill makes an angle of with the horizontal. (a) What force must he exert parallel to the slope? (b) What is the force perpendicular to the surface of the hill that presses the sled against the hill? 335
36 4. A displacement vector makes an angle of with respect to the xaxis. If the ycomponent of the vector is equal to 150 cm, what is the magnitude of the displacement vector? 5. A plane is traveling northeast at 200 km/hr. What is (a) the northward component of its velocity, and (b) the eastward component of its velocity? 6. While taking off, an airplane climbs at an 8 0 angle with respect to the ground. If the aircraft s speed is 200 km/hr, what are the vertical and horizontal components of its velocity? 7. The vector A has a magnitude of 50.0 m and points in a direction of north of east. What are the magnitudes and directions of the vectors, (a) 2A, (b) 0.5A, (c) A, (d) 5A, (e) A + 4A, (f) A 4A. 8. A car that weighs 8900 N is parked on a hill that makes an angle of 43 0 with the horizontal. Find the component of the car s weight parallel to the hill and perpendicular to the hill. 9. A girl pushes a lawn mower with a force of 90 N. The handle of the mower makes an angle of 40 0 with the ground. What are the vertical and horizontal components of this force and what are their physical significances? What effect does raising the handle to 50 0 have? 10. A missile is launched with a speed of 1000 m/s at an angle of 73 0 above the horizontal. What are the horizontal and vertical components of the missile s velocity? 11. When a ladder leans against a smooth wall, the wall exerts a horizontal force F on the ladder, as shown in the diagram. If F is equal to 50 N and θ is equal to 63 0, find the component of the force perpendicular to the ladder and the component parallel to the ladder. Diagram for problem Unit Vectors 12. Express the following three displacements in vector notation in terms of the unit vectors i and j: (a) 3 miles due east, (b) 6 miles eastnortheast, and (c) 7 miles northwest. 13. A 50 N force is directed at an angle of above the horizontal. (a) Resolve this force into vertical and horizontal components, and (b) write the vector in terms of unit vectors If a = 5i + 6j, find the magnitude of a and the angle θ that the vector makes with the xaxis. 15. A point in space is located by the position vector 336
Figure 1.1 Vector A and Vector F
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