1.7.  The Components of a Vector

VECTOR COMPONENTS

Suppose a car moves along a straight line from start to finish in Figure 1.15, the corresponding displacement vector being r. The magnitude and direction of the vector r give the distance and direction traveled along the straight line. However, the car could also arrive at the finish point by first moving due east, turning through 90°, and then moving due north. This alternative path is shown in the drawing and is associated with the two displacement vectors x and y. The vectors x and y are called the x vector component and the y vector component of r.

Figure 1.15  The displacement vector r and its vector components x and y.

Vector components are very important in physics and have two basic features that are apparent in Figure 1.15. One is that the components add together to equal the original vector:

The components x and y, when added vectorially, convey exactly the same meaning as does the original vector r: they indicate how the finish point is displaced relative to the starting point. In general, the components of any vector can be used in place of the vector itself in any calculation where it is convenient to do so. The other feature of vector components that is apparent in Figure 1.15 is that x and y are not just any two vectors that add together to give the original vector r: they are perpendicular vectors. This perpendicularity is a valuable characteristic, as we will soon see.

Any type of vector may be expressed in terms of its components, in a way similar to that illustrated for the displacement vector in Figure 1.15. Figure 1.16 shows an arbitrary vector A and its vector components A_x and A_y. The components are drawn parallel to convenient x and y axes and are perpendicular. They add vectorially to equal the original vector A:

Figure 1.16  An arbitrary vector A and its vector components A x and A y .

There are times when a drawing such as Figure 1.16 is not the most convenient way to represent vector components, and Figure 1.17 presents an alternative method. The disadvantage of this alternative is that the tail-to-head arrangement of A_x and A_y is missing, an arrangement that is a nice reminder that A_x and A_y add together to equal A.

Figure 1.17  This alternative way of drawing the vector A and its vector components is completely equivalent to that shown in Figure 1.16

The definition that follows summarizes the meaning of vector components:

The values calculated for vector components depend on the orientation of the vector relative to the axes used as a reference. Figure 1.18 illustrates this fact for a vector A by showing two sets of axes, one set being rotated clockwise relative to the other. With respect to the black axes, vector A has perpendicular vector components A_x and A_y; with respect to the colored rotated axes, vector A has different vector components and . The choice of which set of components to use is purely a matter of convenience.

Figure 1.18  The vector components of the vector depend on the orientation of the axes used as a reference.

SCALAR COMPONENTS

It is often easier to work with the scalar components, A_x and A_y (note the italic symbols), rather than the vector components A_x and A_y. Scalar components are positive or negative numbers (with units) that are defined as follows. The component A_x has a magnitude that is equal to that of A_x and is given a positive sign if A_x points along the +x axis and a negative sign if A_x points along the –x axis. The component A_y is defined in a similar manner. The following table shows an example of vector and scalar components:

Vector Components	Scalar Components	Unit Vectors
Ax=8 meters, directed along the +x axis	Ax=+8 meters	Ax=(+8 meters)
Ay=10 meters, directed along the –x axis	Ay=–10 meters	Ay=(–10 meters)

In this text, when we use the term “component,” we will be referring to a scalar component, unless otherwise indicated.

Another method of expressing vector components is to use unit vectors. A unit vector is a vector that has a magnitude of 1, but no dimensions. We will use a caret () to distinguish it from other vectors. Thus,

	is a dimensionless unit vector of length l that points in the positive x direction, and
	is a dimensionless unit vector of length l that points in the positive y direction.

These unit vectors are illustrated in Figure 1.19. With the aid of unit vectors, the vector components of an arbitrary vector A can be written as A_x=A_x and A_y=A_y , where A_x and A_y are its scalar components (see the drawing and third column of the table above). The vector A is then written as A=A_x +A_y

Figure 1.19  The dimensionless unit vectors and have magnitudes equal to 1, and they point in the + x and + y directions, respectively. Expressed in terms of unit vectors, the vector components of the vector A are A x and A y .

Two vectors, A and B, are shown in the drawing that follows. (a) What are the signs (+ or –) of the scalar components Ax and Ay of vector A? (b) What are the signs of the scalar components Bx and By of vector B? (c) What are the signs of the scalar components Rx and Ry of the resultant vector R, where R=A+B? Two concepts play a role in this question: the scalar components of a vector, and how two vectors are added by means of the tail-to-head method to produce a resultant vector. For similar questions (including calculational counterparts), consult Self-Assessment Test 1.2. This test is described at the end of Section 1.8.

RESOLVING A VECTOR INTO ITS COMPONENTS

If the magnitude and direction of a vector are known, it is possible to find the components of the vector. The process of finding the components is called “resolving the vector into its components.” As Example 7 illustrates, this process can be carried out with the aid of trigonometry, because the two perpendicular vector components and the original vector form a right triangle.

A displacement vector r has a magnitude of r=175 m and points at an angle of 50.0° relative to the x axis in Figure 1.20. Find the x and y components of this vector. Figure 1.20  The x and y components of the displacement vector r can be found using trigonometry. Reasoning  We will base our solution on the fact that the triangle formed in Figure 1.20 by the vector r and its components x and y is a right triangle. This fact enables us to use the trigonometric sine and cosine functions, as defined in Equations 1.1 and 1.2. Problem solving insight Either acute angle of a right triangle can be used to determine the components of a vector. The choice of angle is a matter of convenience.    Solution 1 The y component can be obtained using the 50.0° angle and Equation 1.1, sin q =y/r: In a similar fashion, the x component can be obtained using the 50.0° angle and Equation 1.2, cos q =x/r:    Solution 2 The angle a in Figure 1.20 can also be used to find the components. Since a+50.0°=90.0°, it follows that a=40.0°. The solution using a yields the same answers as in Solution 1:

Since the vector components and the original vector form a right triangle, the Pythagorean theorem can be applied to check the validity of calculations such as those in Example 7. Thus, with the components obtained in Example 7, the theorem can be used to verify that the magnitude of the original vector is indeed 175 m, as given initially:

It is possible for one of the components of a vector to be zero. This does not mean that the vector itself is zero, however. For a vector to be zero, every vector component must individually be zero. Thus, in two dimensions, saying that A=0 is equivalent to saying that A_x=0 and A_y=0. Or, stated in terms of scalar components, if A=0, then A_x=0 and A_y=0.

Two vectors are equal if, and only if, they have the same magnitude and direction. Thus, if one displacement vector points east and another points north, they are not equal, even if each has the same magnitude of 480 m. In terms of vector components, two vectors, A and B, are equal if, and only if, each vector component of one is equal to the corresponding vector component of the other. In two dimensions, if A=B, then A_x=B_x and A_y=B_y. Alternatively, using scalar components, we write that A_x=B_x and A_y=B_y.

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