Vectors follow an algebra that is motivated by their applications, and we shall use them often. These algebraic operations are described in your book, but they include:
 

In addition to these rules, two different ways to multiply vectors are useful. The first is the scalar or dot product of vectors A and B and is defined to result in a scalar having the value
 

where alpha is the angle between A and B. Why is this operation defined this way? The reason is that it turns out to be useful in describing certain physical quantities-work, for example. The second way to multiply vectors is designed to result in a vector. The vector or cross product is defined to have the magnitude
 

where alpha is again the angle between the two vectors (the smaller of the two possible angles). The vector product is useful in describing rotational motion, for example. Unlike the dot product, the vector product is a vector. The direction of the vector (A crossB) is defined by the so-called right-hand rule. Using the fingers of the right hand pointed in the direction of A, the fingers are rotated into the vector B(remember - the smaller of the two possible angles). The vector (A cross B) is then perpendicular to both A and B and points in the direction of one's thumb. A vector dotted into itself gives the square of the length of the vector. Thus A dot A = A². We also see that if A is perpendicular to B, then A dot B = 0. In contrast, a vector crossed into itself (A cross A) is 0. By the right-hand rule, although the scalar product commutes, the vector product does not. That is to say, although A dot B = B dot A, (A cross B) does not equal (Bcross A). Instead, (A cross B) = - (Bcross A).In this discussion we concentrate on the operations of addition and subtraction. The operation of scalar and vector multiplication will be discussed later in the course, but only to the extent that they are needed.
 

In summary, we completely specify the vector either by writing A in terms of (A, alpha ) or A in terms of (A_x, A_y).

Looking back to Fig. 1, we can draw on the figure any rectangular system whatsoever, then calculate the components of the vectors A and B separately in the system we have drawn. Then, knowing the components of A and B separately, we automatically also know the components of the sum C = A + B in the same system, because
 

The algebra of adding and subtracting vectors becomes clear. We choose a coordinate system, calculate the components in that system and add and subtract the components!

As in the two-dimensional case, this vector is completely specified once we know its components along each of the three axes. In order to express A in terms of the chosen coordinate system, we need to write it in such a a way that its vector nature is clear. Toward this goal, we define three unit vectors (vectors with a magnitude of 1), each parallel to one of the three axes. In our x-y-z system these unit vectors are i, j and k and are parallel to x, y and z, respectively. Now, if the unit vector i is multiplied by the scalar Ax, then Axi is a vector of length Ax and directed parallel to the x axis. As a consequence, A can be written
 

This is handy notation, knowing that adding and subtracting vectors is done using components. The notation gives us an algebraic form in which to express a variety of vector operations. Thus, for example, the difference between two vectors, A and B can be expressed as B - A = (B_x - A_x) i + (B_y - A_y) j + (B_z - A_z) k. For many problems, the full three dimensions may not be needed because the motion is confined to a plane. In this case, the common practice is to use x and y, eliminating z.

As discussed earlier, a general vector describes a physical quantity at a point in space and does not depend on any coordinate system. Coordinate axes can be placed anywhere and with any orientation. When described in terms of different cartesian systems, a given vector may have different looking components, but the length and orientation of the vector remain the same.
 

Let us assume that we want to describe the position of an object as it moves through space along a path described by the function P(x,y,z). We use a rectangular system as before, with orthogonal x-y-z axes. In this coordinate system we draw a special kind of vector called a position vector, with the tail at the origin and the head at the position P(x,y,z). This vector, labeled R in Figure 3, is completely specified by giving the components along each axis. In order to express R in terms of the chosen coordinate system, we use the three unit vectors i, j and k. As a consequence, R is

As the object moves through space, the position vector traces a path. Let us assume that the value of R at two different times is known. The difference between these two position vectors is
 

This last expression is a vector that goes from (x₁, y₁, z₁) to (x₂, y₂, z₂) and is exactly like the vectors A or B discussed in Section 4. An important difference between a position vector R and a general vector such as delta R is that the components of R are x, y and z, whereas for delta R the components are delta x, delta y and delta z. It is important to distinguish between true vectors and position vectors. A true vector does not depend on coordinate system but only on the difference between one end of the vector and the other. A position vector, in contrast, does depend on the coordinate system, because it is used to locate a position relative to a specified reference point.

Some of the previous discussion may feel abstract to you. Do not worry about that because you gradually will become comfortable with the ideas as you use them and as you see then used in lectures. Vector notation is simply a short-hand way of expressing information in a compact but precise way.