Vectors Primer M.. Simani Jul 7, 2007 This note gives a short introduction to the concept of vector and summarizes the basic properties of vectors. Reference textbook: Universit Phsics, Young and Freedman, ddison Wesle Ed. 1
1 Introduction: what is a vector? Phsical quantities can be divided into two categories scalars and vectors. scalar is a quantit which is full described b its magnitude, such as temperature, time, mass, and electric charge. scalar is a number and is alwas positive. vector is an object which is full described b both magnitude ND direction, such as velocit, force, and electromagnetic field. s an example, ou can easil understand that when dealing with a push or pull force, the direction in which the force is acting is as important as the strength or magnitude of the force. Note that some scalar quantities can sometimes be negative, e.g. a temperature of 20. However, temperature is not a vector. Vectors are tpicall represented in bold tpe with an arrow above them: (1) The arrow is a reminder that vectors have direction. The magnitude of vectors is a scalar quantit and is indicated b (no bold tpe and no arrow on top) or (vector smbol with vertical bars). In the remaining of this paper we will omit the bold tpe in the vector notation, but we will alwas place an arrow on top. vector is also drawn as an arrow and the direction of the head of the arrow gives the direction of the vector (Figure 1). The Figure 1: Graphical representation of a vector. length of the arrow in a given scale represents the magnitude of the vector. When more than one vector needs to be drawn, each arrow is drawn with the proper length according to the chosen scale. The arrow points in a precise direction. Directions are described b the use of some convention, such as the definition of a x- reference frame with a specific origin. n example of the use of the concept of vector is the displacement vector because it indicates how (magnitude and direction) an object has been displaced b its starting point. Two vectors are equal when the have the same magnitude and the same direction, no matter where the are located in space. If two vectors have the same direction the are also parallel (Figure 2 (a)). Two vectors that are equal in magnitude but opposite in directions are called antiparallel (Figure 2 (b)). In this wa we can define the negative of a vector as a vector having the same magnitude of but opposite direction, and it is denoted as. 2
= (a) (b) Figure 2: (a) Example of equal vectors; (b) Example of a vector and its corresponding negative vector. 1.1 perations with vectors ddition The sum of two vectors, and, is a vector, which is obtained b placing the tail of the arrow representing on the head of the arrow representing, and then drawing a line from the tail of to the head of (see Figure 3 (a)). This is referred to the Head-to-Tail method. lternativel, ou can translate such that its tail coincides with that of, the sum vector is the diagonal of the parallelogram constructed using and (Figure 3 (b)). ther examples of vector sum are illustrated in Figure 3 (c) and (d). lwas remember that adding two vectors is a geometrical process. Therefore the sum of two vectors is not just the sum of the magnitudes of the vectors, but the constructed direction should also be considered. f course, it is possible to add more than two vectors as follow + + = (2) ( + ) + = D + = R (3) Figure 4 illustrates the procedure using the head-to-tail method (b) and the parallelogram method (c). Subtraction The result of the subtraction operation is a vector, which can be obtained using the same rules of addition for the vectors and. The Head-to-Tail method and the Parallelogram method are illustrated in Figure 5 (b) and (c), respectivel. Multiplication b a scalar The result of the operation a is a vector, which has magnitude a and the same direction as. In general, man of the laws of algebra holds also for vector algebra: ommutative law for addition: + = + 3
(a) (b) + = (c) (d) Figure 3: Examples of addition of vectors: (a) tail-to-head method; (b) parallelogram method; (c) and (d) tail-to-head method, note that in these cases the vectors are collinear. 4
(a) (b) R (c) D D R Figure 4: Examples of addition of more than two vectors. Head-to-Tail method (b) and Parallelogram method (c). (a) = + ( ) = (b) (c) Figure 5: Examples of subtraction of vectors. 5
ssociative law for addition: ( + ) + = + ( + ) ommutative law for multiplication: a = a ssociative law for multiplication: (a + b) = a + b Distributive law: a( + ) = a + a 1.2 omponents of a vector vector can also be defined b its components. In order to do that ou need to specif a artesian coordinate sstem of axes which will define the x -plane and the origin of the sstem. n vector can then be decomposed into the x and components vectors b projecting the vector on the x and axes (see Figure 6). definition, the component vectors of vector lie along a coordinate axis of the artesian sstem and are perpendicular to each other such that = x + (4) The magnitudes of the component vectors indicated b x and (no arrow on top!) are the components of. Note that when the component vector points in the negative direction of the artesian axis (as illustrated in Figure 6 (b) and (c)), its magnitude is still a positive number (the magnitude of a vector is never negative!) but the direction is in this case negative. The minus sign placed next to the component magnitude is a reminder that the component vector points along the negative axis. This needs to be taken into account carefull when composing vectors. The components of the vector can be calculated from its magnitude and its direction as follow x = cos (5) = sin (6) The angle is defined as the angle between the positive x-axis and the vector rotating towards the positive -axis (see Figure 6). Given the x- and -components of the vector, its magnitude and direction can be calculated as follow = 2 x + 2 (7) tan = x = arctan x (8) Note that an two angles that differ b 180 have the same tangent. Draw a sketch of the vector on the artesian sstem and look at the individual components to check which of the two possibilities is the correct one. When adding two or more vectors, we can now use their components to to calculate the vector sum. Figure 7 illustrates the procedure for a simple case in (a) and for a more complicated case in (b). The general rule to find magnitude and 6
(a) (b) x x x x (c) x x (d) omponents! x x ( ) Figure 6: Examples of component vectors (a-c), and components (d). In (d) the magnitude of is a positive number, but I indicated a minus sign next to it to take note of the negative direction. 7
direction of a vector R resulting from the sum R = + is R x = x + x (9) R = + (10) = arctan R R x (11) We find easier to evaluate the vector components and vector directions when all the given vectors in the sum are transposed with their tail to the origin and then appling the parallelogram method as graphical aid (see Figure 7 (c)). In this wa the sign with which the components need to be taken into account comes automaticall from their projection on the artesian axes. lwas draw a sketch of the problem before starting to solve an equation. Then check our results with the sketch to see if it makes sense. lso, keep ourself consistent with the notation for the angle. (a) R x x x (b) (c) Parallelogram method x R x x R x Figure 7: Examples of using components to add vectors. 8
1.3 Unit vectors natural consequence of the vector components described above is to define unit vectors. unit vector is a vector with magnitude of 1 and direction along one of the artesian axis. In an x--z coordinate sstem the three unit vectors are generall indicated as î, ĵ, and ˆk (note the special hat on top) and the represent unit vectors pointing towards the positive direction of the corresponding axis. Figure 8 illustrate an example of decomposition of a vector in a x- plane into its component vectors and its unit vectors. The relationship j^ j^ i^ x i^ x Figure 8: Vector decomposition into component vectors and unit vectors. between component vectors of a vector in the x--z sstem and unit vectors is expressed as follows: x = x î (12) = ĵ (13) z = zˆk (14) Similarl, the vector can be also written as = x î + ĵ + zˆk (15) s a consequence, we can express the vector R resulting from the sum R = + as follows: R = + (16) = ( x î + ĵ + zˆk) + (x î + ĵ + zˆk) (17) = ( x + x )î + ( + )ĵ + ( z + z )ˆk (18) = R x î + R ĵ + R zˆk (19) 9
Note that the breaking up of a vector into its either component vectors or unit vectors is not unique. Depending on the orientation of the coordinate sstem with respect to the vector in question, it is possible to have more than one set of components. This will become relevant when later we will introduce the concept of force. In that case, the right choice of a reference sstem can simplif the solution of the problem. 1.4 Product of vectors The multiplication of two vectors, is not uniquel defined. There are two tpes of vector multiplication: the first one, the scalar product or dot product of two vectors results in a scalar quantit, not a vector; the second, the vector product or cross product of two vectors results in another vector. Scalar product The definition of the scalar product of two vectors is = cos (20) where is the angle between the two vectors. The angle is alwas between 0 and 180. s illustrated in Figure 9, the scalar product is basicall given b the magnitude of multiplied b the component of parallel to (the projection of over ). The result of a scalar cos Figure 9: Graphical representation of the scalar product. Note that the two vectors are drawn with the same starting point. product is a scalar quantit, not a vector, and it ma be positive, negative, or zero, depending on the angle. The product is positive for between 0 and 90, and negative for 90 and 180. In particular, the scalar product of two perpendicular vectors is alwas zero. Note that =, thus the scalar product of two vectors is commutative. In terms of their components, the product follows b expanding the product in terms of unit vectors = (x î + ĵ + zˆk) + (x î + ĵ + zˆk) (21) 10
= x x + z z + z z (22) where we used the fact that î î = ĵ ĵ = ˆk ˆk = 1 and î ĵ = î ˆk = ĵ ˆk = 0 because î, ĵ, and ˆk are all perpendicular to each other. The scalar product gives a straightforward wa to find the angle between an two vectors and whose components are known: = x x + + z z = cos (23) ( ) x x + z z + z z = = arccos (24) The scalar product is used to describe the work done b a force. When a constant force F is applied to a bod that undergoes a displacement s, the work W is a scalar quantit given b W = F S. Vector product For a graphical description of the vector product look at Figure 10. The vectors = x Figure 10: Graphical representation of the vector product. and lie on a plane and their tail are at the same point. We define the vector product as a vector with magnitude = sin (25) and direction perpendicular to the plane where and are laing. In this case, geometricall the magnitude is obtained b multipling the magnitude b the perpendicular component of the vector over. The angle is chosen as the smallest angle ou can get when moving towards. Therefore, the angle is alwas between 0 and 180 and is alwas positive (it is a vector magnitude!). Note that the vector product of two parallel or antiparallel vectors is alwas zero, including the vector product of an vector with itself. Since there are two possible directions for a vector perpendicular to a plane, to define the specific direction of the vector we use the Right-hand rule: put our hand on the vectors plane and align our right hand along the vector in such a wa the the arrow is entering into our hand. 11
Now curl our fingers of our right hand in choosing the smaller of the two possible angles between and. Your thumb is now pointing in the direction of. Note that =, so the vector product is not commutative. If we know the components of and, the components of can be calculated as follow = (26) = ( x î + ĵ + zˆk) (x î + ĵ + zˆk) (27) =... few das later... (28) = ( z z )î + ( z x x z )ĵ + ( x x )ˆk (29) In the derivation above we considered the fact that because all parallel to themselves, and from the right-hand rule î î = ĵ ĵ = ˆk ˆk = 0 (30) î ĵ = ĵ î = ˆk (31) ĵ ˆk = ˆk ĵ = î (32) ˆk î = î ˆk = ĵ (33) (34) because the are all perpendicular to each other. Finall, the vector product can also be expressed in determinant form as = î ĵ ˆk x z x z (35) but this is another stor. The vector product is commonl used to calculate vector quantities like angular momentum and torque. lso, it is extensivel used when combining electric and magnetic fields. 12
2 Trigonometric Table Given the right triangle in Figure 2, we can define the following relations cos = h sin = h tan = = arctan (36) (37) (38) (39) where h is the length hpothenuse of the right triangle, is the length of the side adjacent to the angle, and is the length of the side opposite to the angle. h Figure 11: Right triangle. 13