The motion of a particle in one dimension is simple. Its velocity is either positive or negative: positive velocity corresponds to a motion to the right while negative velocity corresponds to a motion to the left. To describe the motion of an object in 3 dimensions, we need to specify not only the magnitude of its velocity, but also its direction: velocity is a vector. A quantity not involving a direction is a scalar. Examples of scalars are temperature, pressure and time. In this Chapter we will discuss the various vector operations that will be used in this course.
One important vector operation that we will frequently encounter is vector addition. There are two methods for vector addition: the graphical method and the analytical method. We will start discussing vector addition by using the graphical method.
Assume two vectors and are defined. If is added to , a third vector is created (see Figure 3.1).
Figure 3.1. Commutative Law of Vector Addition.
There are however two ways of combining the vectors and (see Figure 3.1). Inspection of the resulting vector shows that vector addition satisfies the commutative law (order of addition does not influence the final result):
Vector addition also satisfies the associative law (the result of vector addition is independent of the order in which the vectors are added, see Figure 3.2):
Figure 3.2. Associative Law of Vector Addition.
Figure 3.3. Vector and -
The opposite of vector is a vector with the same magnitude as but pointing in the opposite direction (see Figure 3.3):
+ (- ) = 0
Subtracting from is the same as adding the opposite of to (see Figure 3.4):
= - = + (- )
Figure 3.4. Vector Subtraction.
Figure 3.4 also shows that + = .
In actual calculations the graphical method is not practical, and the vector algebra is performed on its components (this is the analytical method).
To demonstrate the use of the analytical method of vector addition, we limit ourselves to 2 dimensions. Define a coordinate system with an x-axis and y-axis (see Figure 3.5). We can always find 2 vectors, and , whose vector sum equals . These two vectors, and , are called the components of , and by definition satisfy the following relation:
Suppose that [theta] is the angle between the vector and the x-axis. The 2 components of are defined such that their direction is along the x-axis and y-axis. The length of each of the components can now be easily calculated:
and the vector can be written as:
Figure 3.5. Decomposition of vector .
Note: in contrast to Halliday, Resnick, and Walker we use and to indicate the unit vectors along the x- and y-axis, respectively (Halliday, Resnick, and Walker use i, j, and k which is harder to write). The decomposition of vector into 2 components is not unambiguous. It depends on the choice of the coordinate system (see Figure 3.6).
Although the decomposition of a vector depends on the coordinate system chosen, relations between vectors are not affected by the choice of the coordinate system (for example, if two vectors are perpendicular in one coordinate system, they are perpendicular in every coordinate system). The physics (and the relation between physical quantities) is also not affected by the choice of the coordinate system, and we usually choose the coordinate system such that our problems can be solved most easily. This was already demonstrated in Chapter 2, where the origin of the coordinate system was usually defined as the position of the object at time t = 0. In two or three dimensions, we can usually choose the coordinate system such that one of the vectors coincides with one of the coordinate axes. For example, if the coordinate system is defined such that the direction of is along the x-axis, the components of are:
ax = a
ay = 0
az = 0
Figure 3.6. Decomposition of vector in different coordinate systems.
Vector algebra using the analytical method is based on the following rule:
"Two vectors are equal to each other if their corresponding components are equal"
Applying this rule to vector addition:
Comparing the equations for and , we can conclude that since is equal to , their components are related as follows:
These relations describe vector addition using the analytical technique (and similar relations hold for vector addition in three dimensions).
We have just described how to find the components of a vector if its magnitude and direction are provided. If the components of a vector are provided, we can also calculate its direction and magnitude. Suppose the components of vector along the x-axis and y-axis are ax and ay, respectively. The length of the vector can be easily calculated:
The angle [theta] between the vector and the positive x-axis can be obtained from the following relation (see also Figure 3.5):
Note: one should exercise great care in using the previous relation for [theta]. Atan(ay/ax) has 2 solutions; any calculator will return the solution between - [pi]/2 and [pi]/2. The correct solution will depend on the sign of ax:
if ax > 0: - [pi]/2 < [theta] < [pi]/2
if ax < 0: [pi]/2 < [theta] < 3[pi]/2
Figure 3.7. Example of angle ambiguity.
Example: The two vectors shown in Figure 3.7 can be written as:
It is clear that for both vectors: atan([theta]) = 1. For vector , the solution is [theta] = 45deg., while for vector , the solution is [theta] = 135deg.. Note that ax = 1 > 0, and bx = -1 < 0.
The product of a vector and a scalar s is a new vector, whose direction is the same as that of if s is positive or opposite to that direction if s is negative (see Figure 3.8). The magnitude of the new vector is the magnitude of multiplied by the absolute value of s. This procedure can be summarized as follows:
Figure 3.8. Multiplying a vector by a scalar.
Since two vectors are only equal only if their corresponding components are equal, we obtain the following relation between the components of and the components of :
The following relations are summarizing the relations between the magnitude and direction of the vectors and :
Two vectors and are shown in Figure 3.9. The angle between these two vectors is [phi]. The scalar product of and (represented by . ) is defined as:
In a coordinate system in which the x, y and z-axes are mutual perpendicular, the following relations hold for the scalar product between the various unit vectors:
Suppose that the vectors and are defined as follows:
Figure 3.9. Scalar Product of vectors and .
The scalar product of and can now be rewritten in terms of the scalar product between the unit vectors along the x, y and z-axes:
Note that in deriving this equation, we have applied the following rule:
An alternative derivation of the expression of the scalar product in terms of the components of the two vectors can be easily derived as follows (see Figure 3.10). The components of and are given by:
Figure 3.10. Alternative Derivation of Scalar Product.
ax = a cos(a)
ay = a sin(a)
bx = b cos([beta])
by = b sin([beta])
The scalar product can now be obtained as follows:
What is so useful about the scalar product ? If two vectors are perpendicular, their relative angle equals 90deg., and . = 0. The scalar product therefore provides an easy test to determine whether two vectors are perpendicular. Suppose and are defined as follows:
These two vectors are perpendicular since:
In a similar manner we can easily determine whether the angle between the two vector is less than 90deg. (in which case . > 0) or more than 90deg. (in which case . < 0)
The vector product of two vectors and , written as x , is a third vector with the following properties:
where [phi] is the smallest angle between and . Note: the angle between and is [phi] or 360deg. - [phi]. However, since sin([phi]) = - sin(2[pi] - [phi]), the vector product is different for these two angles.
It is clear from the definition of the vector product that the order of the components is important. It can be shown, by applying the right hand rule, that the following relation holds:
x = - x
The following expression can be used to calculate x if the components of and are provided: