3.1. Vector AdditionOne 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.
3.1.1. The graphical methodAssume 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.
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):
+ (- ) = 0Subtracting 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).
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).
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:
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:
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:
3.2. Multiplying Vectors - Multiplying a Vector by a ScalarThe 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.
3.3. Multiplying Vectors - Scalar ProductTwo 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:
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:
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:
3.4. Multiplying Vectors - Vector ProductThe vector product of two vectors and , written as x , is a third vector with the following properties:
- the magnitude of
is given by:
- The direction of
is perpendicular to the plane defined by
and the direction (up or down) is determined using the right-hand rule.
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 = - xThe following expression can be used to calculate x if the components of and are provided: