Physics Lesson 3.2.3 - Position in the system of coordinates

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Welcome to our Physics lesson on Position in the system of coordinates, this is the third lesson of our suite of physics lessons covering the topic of Position, Reference Point, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Position in the system of coordinates

It is known that in a system of coordinates we can assign a letter to each direction available. Thus, if there is only one direction available (1-D) as shown in the above figure, we denote the axis by the letter x and the object's position by the vector x⃗ or Ox)⃗. We can write

and

When two directions of motion (2-D) are available, we can express the position of an object using a pair or coordinates (one for each direction). We must use two letters (usually x and y) to label the directions. Therefore, we need to know both coordinates to determine the location of an object in 2-D. Look at the figure below:

From the above graph, we can see that the object A is 4m on the right and 2m above the reference point. Therefore, we say the position of the object A is at (4m, 2m) and by this, we understand that the position of object A is represented through the vector

and the object A is

away from the origin in the direction of the vector OA⃗.

Likewise, we can use the same approaches in 3-D (in space) as well. We have another direction added in this case. Usually, it is denoted by the letter z. Therefore, we must write all three coordinates to determine the position of an object. Look at the graph below.

The object A is in the 3 dimensional space. It is diverted 6m from the origin according the x-direction, 5 m from the origin in the y-direction and 6m from the origin in the z-direction, all in the positive direction. This means the components of the vector OA⃗ which represents geometrically the linear distance from the origin, are OAx = 6m, OAy = 5m and OAz = 6m respectively.

From the concept of vector's magnitude, we know that

Therefore, substituting the values, we obtain for the magnitude of the vector OA⃗

Thus, the object A is nearly 10m away from the origin (reference point) in the direction of the vector OA⃗.

Example 1

Write the position of the objects A, B and C shown in the figure. How far are they from the origin?

Solution 1

The object A is in the xOy plane. It has no z-coordinate, so we need to know only two coordinates to show its position.

From the figure, we can see that Ax = 4m, Ay = 3m (and Az = 0m). Therefore, the position OA⃗ of the object A is

The distance from the origin of the Object A is found by calculating the magnitude of the vector OA⃗. Therefore, using the known procedure explained in the article "Vectors and Scalars" for calculating the magnitude of a vector, we can write

Substituting the values, we obtain for the magnitude of the vector OA⃗

This means the point A is 5m away from the origin in the direction of the vector OA⃗.

The same procedure is used for the other two objects. Thus, for the object B we have only one coordinate Bz = 5m as it lies on the z-axis only. It is not necessary to calculate the magnitude of the vector OB⃗ as it is clear that |OB⃗|= 5m.

As for the object C, we can see from the figure that it contains all three coordinates. Thus, Cx = 3m, Cy = 6m and Cz = 4m. Therefore, the magnitude of the vector OC⃗ which represents the position of the object C (its linear distance from the origin), is

Substituting the values, we obtain for the magnitude of the vector (OC)⃗

Therefore, the object C is 7.81m away from the origin in the direction of the vector OC⃗.

The figure below shows the position vectors for the three objects.

Remark! The 1 and 2 dimensional motions are special cases of the 3 dimensional motion. We can either write 0 in the place of the missing coordinates or simply represent the position in as many coordinates as given. We can illustrate this aspect using the position of the vector B. This position can be mathematically represented in three ways:

In one dimension (according z only)

In two dimensions (according x and z, or y and z)

In three dimensions (according x, y and z)

All these three presentations show the same thing: the vector OB⃗. Hence, they are all equivalent.

You have reach the end of Physics lesson 3.2.3 Position in the system of coordinates. There are 3 lessons in this physics tutorial covering Position, Reference Point, you can access all the lessons from this tutorial below.

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