A particle confined to a straight line can move in only two directions. We can take its motion to be positive in one of these directions and negative in the other. For a particle moving in three dimensions, however, a plus sign or minus sign is no longer enough to indicate the direction of the motion. Instead, we must use a vector.
A vector has magnitude as well as direction, and vectors follow certain rules of combination, which we examine below; a vector quantity is a quantity that has both a magnitude and a direction. Some physical quantities that can be represented by vectors are displacement, velocity, and acceleration.
Not all physical quantities involve a direction. Temperature, pressure, energy, mass, and time, for example, do not “point” in the spatial sense. We call such quantities scalars, and we deal with them by the rules of ordinary algebra. A single value, with a sign (as in -40ºF), specifies a scalar.
The simplest vector quantity is displacement, or change of position. A vector that represents a displacement is called, reasonably, a displacement vector. (Similarly, we have velocity vectors and acceleration vectors.) If a particle changes its position by moving from A to B in Figure v-1. we say that it undergoes a displacement from A to B, which we represent with an arrow pointing from to B. The arrow specifies the vector graphically. To distinguish vector symbols from other kinds of arrows, we use the outline of a triangle as the arrowhead.
The arrows from A to B, from A’ to B’ represent the same change of position for the particle and we make no distinction among them. All three arrows have the same magnitude and direction and thus specify identical displacement vectors.
ADDING VECTORS: GRAPHICAL METHOD
Suppose that, as in the vectors diagram Figure v-1, a particle moves from A to B and then later from B to C. We can represent its overall displacement (no matter what its actual path) with two successive displacement vectors, AB and BC. The net effect of these two displacements is a single displacement from A to C. We call AC the vector sum (or resultant) of the vectors AB and BC. This sum is not the usual algebraic sum, and we need more than simple numbers to specify it.
In Figure B, we redraw the vectors of Figure A and relabel them in the way that we shall use from now on, namely, with boldface symbols such as a, b, and s. In handwriting, you can place an arrow over the symbol, as in ā. If we want to indicate only the magnitude of the vector ( a quantity that is always positive), we shall use the italic symbol, such as a, b, and s. (You can use just a handwritten symbol.) The boldface symbol always implies both properties of the vectors, magnitude and direction.
We can represent the relation among the three vectors in Figure v-2 with the vector equation.
s = a + b, ( Eq. 1 )
which says that the vector s is the vector sum of vectors a and b. The symbol + in Eq. 1 and the words “sum” and “add” have different meanings for vectors than they do in the usual algebra.
Figure A and B suggests a procedure for adding two-dimensional vectors a and b graphically. (1) On a sheet of paper, lay out vector a to some convenient scale and at the proper angle. (2) Lay out vector b to the same scale, with its tail at the head of a, again at the proper angle. (3) Construct the vector sum s by drawing an arrow from the tail of a to the head of b. Note that this procedure takes into account both the magnitude and the directions of the vectors; you can easily generalize it to add more than two vectors.
Vector addition, defined in this way, has two important properties. First, the order of addition does not matter. That is,
a + b = b + a ( commutative law) Eq. 2
Figure v-3 should convince you that this is the case.
Second, if there are more than two vectors, it does not matter how we group them as we add them. Thus if we want to add vectors a, b, and c, we can add a and b first and then add their vector sum to c. On the other hand, we can add b and c first, and then add that sum to a. We get the same result either way. In equation form,
( a + b ) + c = a + ( b + c ) ( associative law ) Eq. 3
The vector - b is a vector with the same magnitude as b but the opposite direction Figure D. If you try to add the two vectors in Figure E, you will see that
b + ( - b ) = 0
Adding – b has the effect of subtracting b. We use this property to define the difference between two vectors: let d = a – b. Then
d = a – b = a + ( - b ) (subtraction) Eq. 4
That is, we find the difference vector d by adding the vector – b to the vector a. Figure v-5 shows how this is done graphically.
Remember, although we have used displacement vectors as a prototype, the rules for addition and subtraction hold for vectors of all kinds, whether they represent forces, velocities, or anything else. However, as in ordinary arithmetic, it is still true that we can add only vectors of the same kind. We can add two displacement, for example, or
Two velocities, but it makes no sense to add a displacement and a velocity. In the world of scalars, that would be like trying to add 21 s and 12 m.
VECTORS AND THEIR COMPONENTS
Adding vectors graphically can be tedious. A neater and easier technique involves algebra but requires that the vectors be placed on a rectangular coordinate system. The x and y axes are usually drawn in the plane of the page, as in Fig. v-6a. The z axis comes directly out of the page at the origin; we ignore it for now and deal only with two-dimensional vectors.
Vector a in Fig. V-6 is in the xy plane. If we drop perpendicular line from the ends of a to the coordinate axes, the quantities ax and ay so formed are called the components of the vector a in the x and y directions. The process of forming them is called resolving the vector. In general, a vector will have three components, although for the case of Fig. V-6a the component along the z axis is zero. As Fig. V-6b shows, if you move a vector in such a way that it remains parallel to its original direction at all time, the values of its component remain unchanged.
We can easily find the components of a Fig. V-6a from the right triangle there:
ax = a cos θ and ay = a sin θ ( Eq. 5 )
where θ is the angle that the vector a makes with the direction of increasing x, and a is its magnitude. Fig. V-6c shows that the vector and its x and y components form a right triangle. It also shows how we can reconstruct a vector from its components: we arrange the components head to tail. Then we complete a right triangle with the vector being along the hypotenuse, from the tail of one component to the head of the other component.
Depending on the value of θ, the components of a vector may be positive, negative, or zero. In a figure, we use small, solid triangle as arrowheads to indicate the signs of the components, according to the usual convention: positive in the direction of increasing coordinate values, and negative in the opposite direction.
Once a vector has been resolved into its components, the components themselves can be used in place of the vector. For example, the two numbers a and θ specify the magnitude and direction of the two-dimensional vector a in Fig. V-6a. But instead of a and θ, we can specify the vector with the two other numbers ax and ay. Both sets of numbers contain exactly the same information, and we can pass back and forth readily between the two description. To obtain a and θ if we are given ax and ay, we note that
Eq. 6In solving problems, you may use either the ax, ay notation or the a, θ notation.
UNIT VECTORS
A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction. It lacks both dimension and unit. Its role purpose is to pint, that is, to specify a direction. The unit vectors in the positive directions of the x, y, and z axes are labeled i, j, and k, as shown in Fig. V-7 is said to be a right-handed coordinate system. The system remains right-handed if it is rotated rigidly to a new orientation. We use such coordinate systems exclusively in this text.
Unit vectors are very useful for expressing other vectors; for example, we can express a and b of Fig. V-6 as
a = axi + ayj
and b = bxi + byj
The quantities axi and ayj are the vector components of a, in contrast to ax and ay, which are its scalar components (or, as before, simply its components).
ADDING VECTORS BY COMPONENTS
As we have noted, adding vectors graphically is tedious; it also has limited accuracy and is challenging in three dimensions. Here we find a more direct technique – adding vectors by combining their components, axis by axis.
To start, consider the statement
r = a + b, ( Eq. 9 )
which says that the vector r is the same as the vector ( a + b ). If that is so, then each component of r must be the same as the corresponding component of ( a + b ):
rx = ax + bx, ( Eq. 10 )
ry = ay + by, ( Eq. 11 )
rz = az + bz, ( Eq. 12 )
In other words, two vectors are equal if their corresponding components are equal. Eq. 10-12 tell us that to add vectors a and b, we must (1) resolve the vectors into their scalar components; (2) axis by axis, combine these scalar components to get the components of the sum r; and (3) if necessary, combine the components of r to get r itself. We have a choice in step 3. We can express r in unit-vector notation, or we can give the magnitude of r and its direction (by stating one angle when we are working in two dimensions of two angles when we are working in three dimensions).
VECTORS AND THE LAWS OF PHYSICS
So far, in every figure that includes a coordinate system, the x and y axes are parallel to the edges of the book page. And so, when a vector a is included, its components ax and ay are also parallel to the edges ( as in Fig. V-8a). The only reason for that orientation of the axes is that it looks proper: there is no deeper reason. We could, instead, rotate the axes (but not the vector a) through an angle φ as in Fig. V-8b, in which case the components would have new values, call them a’x and a’y. Since there are an infinite number of choices of φ, there are an infinite number of different pairs of components for a.
Which then is the “right’ pair of components? The answer is that they are all equally valid because each pair (with its axes) just gives us a different way of describing the same vector a; all produce the same magnitude and direction for the same vector. In Fig. V-8 we have
Eq. 14The point is that we have great freedom in choosing a coordinate system, because the relations among vectors (including, for example, the vector addition of Eq. 1) do not depend on the location of the origin of the coordinate system or on the orientation of the axes. This is also true of the relations of physics; they are all independent of the choice of coordinate system. Add to that the simplicity and richness of the language of vectors and you can see why the laws of physics are almost always presented in that language: one equation, like Eq. 9, can represent three (or even more) relations, like Eq. 10, 11, and 12.
MULTIPLYING VECTORS
There are three ways in which vectors can be multiplied. None is exactly like the usual algebraic multiplication.
Multiplying a Vector by a Scalar
If we multiply a vector a by a scalar s, we get a new vector. Its magnitude is the product of the magnitude of a and the absolute value of s. Its direction is the direction of a it s is positive, but the opposite direction if s is negative. To divide a by s, we multiply a by 1/s.
In either multiplication or division, the scalar may be a pure number or a physical quantity; in the latter case, the physical nature of the product differs from that of the original vector a.
The Scalar Product
There are two ways to multiply a vector by a vector: one way produces a scalar and the other produces a new vector. Students commonly confuse the two ways, and so starting now, you should carefully distinguish between them.
The scalar product of the vectors a and b in Fig.v-9a is written as a.b and defined to be
a.b = ab cos φ ( Eq. 15 )
where a is the magnitude of a, b is the magnitude of b, and φ is the angle * between a and b. Note that there are only scalars on the right side of Eq. 15 (including the value of cos φ). Thus a.b on the left side represents a scalr quantity; because of the notation, it is also known as the dot product and is spoken as “a dot b.”
A dot product can be regarded as the product of two quantities: (1) the magnitude of one of the vectors and (2) the scalar component of the second vector along the direction of the first vector. For example, in Fig.v-9b, a has a scalar component a cos φ along the direction of b; note that a perpendicular dropped from the head of a to b determines that component. Similarly, b has a scalar component b cos φ along the direction of a. If φ is 0º, the component of one vector along the other is maximum, and so also is the dot product. If, instead, φ is 90 º, the component of one vector along the other is zero, and os is the dot product.
Eq. 15 can be rewritten as follows to emphasize the components:
a.b = ( a cos φ ) ( b ) = ( a ) ( b cos φ ) ( Eq. 16 )
The commutative law applies to a scalar product, so we can write
a . b = b . a
When two vectors are in unit-vector notation, we write their dot product as
a . b = (axi + ayj + azk ) . ( bxi + byj + bzk ) ( Eq. 17 )
for which the distributive law applies: each component of the first vector is to be dotted into each component of the second vector.
