ELEMENTS OF VECTOR ALGEBRA


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1 ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions identification and the number of units are often needed to convey adequately the desired information. For instance, to specify fully the velocity of a particle we must give: a. The magnitude of the secondary dimensions, by stating the number of scale units (m/s or km/h). b. The direction of the motion relative to some convenient reference. We have thus far considered only part a. To represent b we use an arrow to indicate direction and relate its length to the magnitude of the quantity involved. Certain quantities having magnitude and direction combine their effects in a special way. Thus the combined effect of two forces acting on a particle, as shown in Fig. A.1, corresponds to the diagonal of a parallelogram formed by the graphical representation of the forces. That is, the quantities add according to the parallelogram law. All quantities that have magnitude and direction and that add according to the parallelogram law are called vector quantities. Other quantities that have only magnitude, such as temperature and work, are called scalar quantities. A vector quantity will be denoted with a boldface letter, which in case of force becomes F. The student may ask: don t all quantities having magnitude and direction combine according to the parallelogram law and therefore become vector quantities? No, not all of them do. One very important example will be pointed out after we consider angular rotation. In the construction of the parallelogram it matters not which force is laid out first. In other words, F 1 combined with F 2 gives the same result as F 2 combined with F 1. In short, the combination is commutative. If the combination is not commutative, then it cannot in general be represented by a parallelogram operation and is thus not a vector. With this in mind, consider the angle of rotation of a body about some axis. We can associate
2 a magnitude (degrees or radians) and a direction (the axis and a stipulation of clockwise or counterclockwise) with this quantity. However, the angle of rotation cannot be considered a vector because in general two rotations about different axes cannot be replaced by a single rotation consistent with the parallelogram law. A.2. MAGNITUDE AND MULTIPLICATION OF A VECTOR BY A SCALAR The magnitude of a quantity, in strict mathematical parlance, is always a positive number of units whose value corresponds to the numerical measure of the quantity. Thus the magnitude of a quantity of measure 50 units is +50 units. The mathematical symbol for indicating the magnitude of a quantity is a set of vertical lines enclosing the quantity. Hence, the above verbal statement can be given mathematically as: 50 units = + 50 units Similarly, the magnitude of a vector quantity is a positive number of units corresponding to the length of the vector in those units. Using our vector symbols we can represent this as: magnitude of a vector A = A Thus A is a scalar quantity. We may now discuss the multiplication of a vector by a scalar. The definition of the product of A by scalar m, written simply as ma, is given as follows: ma is a vector having the same direction as A and the magnitude equal to the ordinary scalar product between magnitudes of m and A. If m is negative, it means simply that vector ma has a direction directly opposite to that of A. The vector A may be considered as the product of the scalar 1 and the vector A. Thus from the above statement we see that A differs from A in that it has an opposite sense. Furthermore, these operations have nothing to do with the line of
3 action of a vector, so A and A may have different lines of action (Fig A.2). This will be the case of the couple to be studied later. A.3. ADDITION AND SUBTRACTION OF VECTORS In adding a number of vectors, we may repeatedly employ the parallelogram construction. We can do this graphically by scaling the lengths of the arrows according to the magnitudes of the vector quantities they represent. The length of final arrow can be interpreted in terms of its length by employing the chosen scale factor. As an example, consider the coplanar vectors A, B, C shown in Fig. A.3. The addition of the vectors can be accomplished in two ways. (to be presented at the lecture). Note that the final vector is identical for both procedures. Thus: A + (B + C) = (A + B) + C When the quantities involved in an algebraic operation can be grouped without restriction, the operation is said to be associative. Thus, the addition of vectors is both commutative, as explained earlier, and associative. We may also add vectors by laying off the vectors head to tail. If many vectors/forces are involved, a polygon will be formed, and the sum of the vectors is the final closing side of the polygon (Fig.A.4.) The process of subtraction is defined in the following manner: to subtract vector B from vector A we reverse the direction of B (i.e., multiply by 1) and then add this new vector to A (Fig.A.5). A.4. RESOLUTION OF VECTORS; SCALAR COMPONENTS The reverse action of addition is called resolution. Thus for a given vector C, we may find a pair of vectors in any two stipulated directions coplanar with C such that the two component vectors sum to the original vector. This can again be accomplished by graphical construction or by using simple helpful sketches and
4 then employing trigonometric relations. (an example will be presented at the lecture). It is also readily possible to find three components not in the same plane as C by the preceding argument. Consider the specification of three orthogonal directions for the resolution of C as is shown in Fig.A.6. It is clear that the vectors C 1, C 2 and C 3 add up to the vector C and are hence called the orthogonal component vectors. The direction of a vector C relative to an orthogonal reference is given by the cosines of the angles formed by the vector and the coordinate axes. These are called direction cosines and are denoted as: cos(c,x) = cos α = l, cos(c,y) = cos β = m, (A.1) cos(c,z) = cos γ = n where α, β, and γ are associated with the x, y, and z axes respectively. It becomes clear from trigonometric considerations that C x = C 1 = C cosα = C l C y = C 2 = C cosα = C m (A.2) C z = C 3 = C cosα = C n Note that the orthogonal scalar components C x, C y and C z may be negative depending on the direction cosines. Using the Pythagorean theorem we obtain: / 2 C = [ C + C + C ] (A.3) x y z Sometimes only one of the vector components described above is desired. Then just one direction is prescribed, as shown in Fig.A.7. Thus the magnitude of the vector component C n is C cosδ. It is obvious that the triangle formed by the vector and its projection is a right triangle. In establishing C n we speak, therefore, of dropping a perpendicular from C to n.
5 A.5. UNIT VECTORS In describing vectors it is sometimes convenient to express them as the product of a scalar times a vector. To do this we use the unit vector, which has a magnitude of unity and a certain prescribed direction. It has no dimensions. To develop such a unit vector in the direction of the vector C, we may formulate the unit vector a so that C a (unit vector in direction C) = (A.4) C We can then express the vector C in the following form: C = C a (A.5) Unit vectors that are of particular use are those directed along the coordinate axes of an orthogonal reference, where i, j, and k correspond to the x, y and z directions as shown in Fig.A.8. Now the vector C from the Fig.A.6 can be expressed as follows C = C x i + C y j + C z k (A.6) A.6. SCALAR OR DOT PRODUCT OF TWO VECTORS In elementary physics, work was defined as the product of the force component in the direction of a displacement times the displacement. In effect, two vectors, force and displacement, are employed to give a scalar, work. In other physical problems, vectors are associated in this same manner so as to result in a scalar quantity. A vector operation that represents such operations concisely is the scalar product (or dot product), which for vectors A and B (see Fig. A.9) is defined as: A B = A B cosα (A.7) where α is the smaller angle between the two vectors. From the definition, it is clear that the dot product is commutative, since the number A B cosα is independent of the order of multiplication of its terms. Thus:
6 A B = B A (A.8) Let us consider A (B + C). By definition, we may project the vector (B + C) onto the direction of A and then multiply the magnitudes. However, the of the sum of two vectors is the same as the sum of the projections of the vectors, which means that: A (B + C) = A B + A C (A.9) An operation on a sum of quantities, which is the same as the sum of the operations on the quantities, is called a distributive operation. Thus the dot product is distributive. If a vector is multiplied by itself to produce a dot product, it forms the square of a number in the following manner: A A = A A = A 2 (A.10) If we express the vectors A and B in Cartesian components when taking the dot product, we get: A B = (A x i + A y j + A z k) (B x i + B y j + B z k) = A x B x + A y B y + A z B z (A.11) Thus, we see that a scalar product of two vectors is the sum of the ordinary products of the respective components. The dot product may be of immediate use in expressing the component of a vector along a given direction as discussed in Sec.A.4. If you refer back to Fig.A.7, you will recall that the component of C along the direction n is given as: C n = C cosδ (Α.12) Now let us consider a unit vector n along the direction of line n. If we carry out the dot product of C and n according to our fundamental definition, the result is: C n = C n cosδ (A.13) But since n is unity, when we compare the preceding equations it is apparent that: C n = C n (A.14) Similarly it is clear that the following useful relations are valid: C x = C i C y = C j C z = C k (A.15)
7 A.7. CROSS PRODUCT OF TWO VECTORS There are other interactions between the vector quantities that represent physical phenomena which result in vector quantities. One such interaction is the moment of a force (to be studied in one of next chapters). To set up a convenient operation for these situations, the vector cross product has been established. For the two vectors (having possibly different dimensions) shown in Fig.A.10 as A and B, the operation is defined as: A B = C (A.16) where C has a magnitude that is given as: C = A B sinα (Α.17) and a direction normal to the plane of AB. The sense, furthermore, corresponds to the direction of advanced for a righthand screw rotated about C as an axis in the direction from A to B that is, from the first stated vector to the second stated vector through the smaller angle between them. In this case, the screw would advance upward in rotating from A to B, whether the procedure is viewed from above or below the plane AB. The description of vector C is now completed, since the magnitude and direction are fully established. The line of action of C in not determined by the cross product, for it depends on the situation. For this product the commutative law breaks down. We can verify, by considering the definition of cross product, that (Α B) = (B A) (A.18) As an exercise, you can demonstrate that the distributive law is still valid for this new operation. Thus A (B + C) = (A B) + (A C) (A.19) Next, consider the cross product of unit vectors. Here the product of equal vectors is zero because α and, consequently, sinα are zero. The product of i j is unity in magnitude and must have direction parallel to the z axis. If the z axis has been erected in a sense consistent with the righthand screw rule when rotating
8 from the x to the y direction, the reference is called a righthand triad, and we can write i j = k In this text, we will use a righthand triad as a reference (Fig.A.11). For ease in evaluating unit cross products for such references, a simple permutation scheme is helpful. In Fig.A.12 the unit vectors i, j, and k are indicated on a circle in a clockwise sequence. Any cross product of a pair of unit vectors results in a positive third unit vector if going from the first vector to the second vector involves a clockwise motion on this circle. Otherwise the vector is negative. Thus: k j = i, k i = j, etc. Next, it will be useful to carry out the cross product in terms of Cartesian components. Using the results of preceding discussions, we get: Α B = (A x i + A y j + A z k) (B x i + B y j + B z k) = (A y B z A z B y )i + (A z B x A x B z )j + (A x B y A y B x )k (A.20) Another method of carrying out this long computation is to evaluate the following determinant: A x B i x A B j y y A B k z z (A.21) Bibliography Beer F.P, Johnston E.R., Jr., Vector Mechanics for Engineers, McGrawHill Shames I.H., Engineering Mechanics Statics, PrenticeHall, 1959
Figure 1.1 Vector A and Vector F
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