physics-online.info: Positions in two and three-dimensional spaces, and position vectors

Topic 1-7: Positions in two and three-dimensional spaces, and position vectors

To specify the position of a point in a two dimensional space, a reference point called the "origin", denoted as 0, and x and y axes need to be specified as in the following picture:

In the left-hand graph the position of point P is defined by its distance to the x-axis, p_y, and its distance to the y-axis, p_x. In the right-hand graph the position of point P is defined by the polar coordinates r and θ, where r is the distance from 0 to P and &theta is the angle between the line 0P and x-axis. In both cases we draw an arrow originating from 0 and ending at point P, and call it the position vector p . In the polar coordinate representation r becomes the length of the position vector and θ the angle it forms with x-axis. In the rectangular coordinate representation p_x and p_y are the projections of the position vector onto x and y axes respectively. Thus we can write

         p=(p_x, p_y)=(r, θ).

In mathematics a vector is defined as an entity that has the magnitude and the orientation in the space, but the originating point of the vector is considered to be irrelevant. Thus in the following picture two equal length and parallel vectors A and B are considered to be identical. However, as position vectors A is clearly different from B since these two position

vectors originate from different reference points and specify different points in the two dimensional space.

Let
        A=(x_a, y_a) and B=(x_b, y_b)
be the rectangular coordinate representations of two position vectors of the same class, that is, they originate from the same reference point. The sum, the subtraction and the scalar products of two position vectors are defined as
        A + B = (x_a+x_b, y_a+y_b),
        A - B = (x_a-x_b, y_a-y_b), (1)
        A·B = x_a·x_b + y_a·y_b.
The following pictures explain the geometric meanings of the sum and the subtraction of two position vectors as defined in Eq. (1).

In the left-hand picture the geometric meaning of the sum A+B is illustrated; first form a parallelogram with position vectors A and B as two sides, then the diagonal is A+B. To verify this procedure, note that the length of the projection of position vector B on x-axis is x_b, and the length of the projection of the upper blue line, that is parallel to B, on x-axis is also x_b. Thus the length of the projection of A+B on x-axis is x_a+x_b. Similarly the length of the projection of A+B on y-axis is y_a+y_b. In the right-hand picture, a parallelogram is formed with B as one side and A as a diagonal; the other side of the parallelogram is A-B. The proof is similar to the case of the sum A+B.

Though the summation and the subtraction between same class position vectors that originate from a common reference point as defined above is elegant, the physical meaning is not very clear. For example, what does it mean to sum two position vectors both of which originate from a common reference point? There is another type of summation that has a clearer physical meaning as illustrated in the following picture.

In the left-hand picture a position vector A originates from the origin 0 and specifies a point R. Another position vector B originates from point R and specifies the position of point P. We define the sum C = A + B as a position vector C that originates from the origin 0 and specify the position of point P. It should be noted that the position vector C has the same rectangular coordinate representation and the same polar coordinate representation as the sum A+B in Eq. (1). However, the sum defined here has a clear physical meaning. Suppose a person, standing at point R, walks a distance equal to the magnitude of the position vector B along the direction of B, then the person will arive at the point P specified by the position vector C = A + B, but no such meaning can be attached to the summation defined by Eq. (1). Clearly position vector B specified here and position vector B specified in Eq. (1) are two totally different type of position vectors and the sum of position vectors here and the sum in Eq. (1) are tow totally different operations, but unfortunately carry the same name due to the confusion of many physicists. Some text books try to consider only the second type of summation as the summation of all vectors and assign no meaning to the definition of Eq. (1). However, this approach also encounters difficulty since there will be different type of vectors, like force vector, that we will come across later, that Eq. (1) type definition will have physical meaning but not the second type of definition of summation of position vectors. It should be noted that in the right-hand picture of the above figure, if the position vector B is translated to originate from an arbitrary point E without changing its orientation, then the sum A+B is neither defined mathematically as in Eq. (1) nor has any physical meaning as in the second type of definition, so it is not defined at all. The cofusion about vectors in physics comes from the reality that there is really no one abstract class of entity called "vector". Every type of vector encountered in physics has vastly different properties and should be considered a different type of entity of its own. Thus, each type of vectors has its own definition of addition and subtraction, as we will see as things develop.

The difference between the two types of definitions of addition and subtraction can be made even clearer when we consider subtraction. In the following picture the result of the subtraction A-B of the second type is illustrated.

Position vectors A and B belongs to the class that originates from the origin, but the result of the subtraction is a position vector that originates from point Q and specifies the point P. This should be contrasted with the subtraction defined according to Eq. (1) and the resulting subtraction A-B that specifies a totally different point as illustrated in an earlier picture.

The extension of position vectors to three dimensional space is obvious. A position vector A that originates from the origin 0 can be written as A = (x, y, z) in the rectangular coordinate representation, or A = (r, θ, φ) in the polar coordinate representation and is illustrated in the following picture:

The definitions of the sum, the subtraction and the scalar product of two position vectors in the three-dimensional space, according to Eq. (1) are
       A = (x_a, y_a, z_a),
       B = (x_b, y_b, z_b),
       A + B = (x_a+x_b, y_a+y_b, z_a+z_b),
       A - B = (x_a-x_b, y_a-y_b, z_a-z_b),
       A•B = x_a•x_b + y_a•y_b + z_a•z_b.
The geometric meanings of the summation and the subtraction of two position vectors are the same as in the two-dimensional case, since when A and B both originate from the origin, they can be drawn in a single plane. Then we define the plane as the x-y plane and everything said about the case of two dimensional space applies equally here. When we deal with the definition of summation and subtraction of the second type, we should also note that all the position vectors involved lie in a single plane. Thus, everything said about the case of two-dimensional space applies here too.

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