**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

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

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

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

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

The geometric meanings of the summation and the subtraction of two position vectors are the same as in the two-dimensional case, since when

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