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We are all familiar with graphs; we know how to draw a line from the origin to the other end until we reach our destination. All of these needs are accomplished through the use of the coordinate system. As a result, the point at which our line marked by an arrow comes to an end is the location at which this ray begins. Think about it this way: you started from home to reach your favourite destination and then routed to another destination, so your arrow changes both in length and direction, indicating that your position vector is changing, and if you choose the shortest path, i.e., displacement, you represent it by a vector called the displacement vector.
Introduction to Position Vector and Displacement Vector
The ideas of position and displacement are fundamental concepts in the study of our physical environment, and they serve as the cornerstone issues for the chapter on the motion. In Euclidean spaces or geometry, the concept of ‘position vector’ has been adopted. The concept of or ‘position vector’ is also referred to as location vector or radius vector. The position of any point in space is described in terms of three coordinates, namely the distances between any arbitrary point marked as “O” or the origin, and the points themselves, namely the distances between any two points denoted as “x,” “y,” and “z.” The straight line between the origin and the point is indicated by the letters ‘r’ or ‘s’. In physics, this vector is used to describe an object at rest or in motion in space in relation to another object, and it is called the reference vector. In accordance with the various locations at various points in time, the vector’s length and direction change in a corresponding manner.
Three dimensions are also used to refer to the three coordinates provided by each direction’s vectors of each direction. Any change in any of these vectors is referred to as displacement. Displacement is defined as any movement of an object from one location to another location along a straight route in everyday language. If anything does not travel in a straight line, the total distance travelled by the object is recorded as the distance travelled. In this scenario, the displacement would be the straight distance between the beginning position and the finishing point. If you’re describing a displacement, which tells you the shortest distance between two spots, it’s also crucial to describe the direction of the displacement so that you can figure out where the final point is exactly. Thus, when we represent the direction of a point, we are referring to it as a position vector, and when we denote the direction of a displacement, we are referring to it as a displacement vector, respectively.
Definition of a Position Vector
When it comes to position vectors, they are described as vectors that signify either the position or the location of any given point in relation to any arbitrary reference point such as the origin. The direction of a position vector is always pointing away from the origin of the vector and towards a specific location.
Example of a Position Vector
In general, the position vector of an object is measured in relation to the origin of the vector. Consider the following scenario: an object is placed in the space as shown:
The Formula for the Position Vector
In order to get the position vector of any point in the xy-plane, we must first determine the coordinates of the point in question. Consider two points A and B, whose coordinates are (x1, y1) and (x2, y2), respectively, and whose coordinates are (x1, y2). In order to get the position vector, we must subtract the respective components of A from B in the following manner:
AB = (x2 – x1, y2 – y1) = (x2 – x1) i + (y2 – y1) j
The position vector AB is drawn from point A to point B, with the origin being point A.
Displacement Vector
The displacement vector of an object is the vector that represents the change in the item’s position vector. Allowing for the possibility that an object is at location A at time = 0 and at point B at time = t. Following are the coordinates of the item at points A and B, along with their respective position vectors:
Position vector at point
^ ^ ^ ^
A= rA=5i+3j+4k
Position vector at point
^ ^ ^ ^
B= rB=2i+2j+1k
Now, the displacement vector of the object from time interval 0 to t will be:
^ ^ ^ ^ ^
rB-rA=-3i+-1j-3k
When an object moves, the vector distance between its initial location and its end point can alternatively be described as the displacement of the object. Consider the following scenario: an object goes from point A to point B along the path depicted by the black curve:
The particle’s displacement would be represented by the vector line AB, which would be travelling in the direction A to B. The displacement vector is constantly travelling in the same direction, from the initial position to the final location.
Things to Remember
- The magnitude of displacement is either less than or equal to the length of an object’s route between two places when it moves between them.
- Unlike magnitudes, vectors can have negative values, but only positive values are possible for vectors. As a result, while scalars are always positive, vectors can be either positive or negative.
- The displacement of an object can be zero if the object is back at its origin, but the distance between two objects can never be zero for an object that has moved.
- Vector operations are distinct from typical algebraic operations in that they involve vectors. They adhere to the triangle law of addition or the parallelogram law of addition, which calculates not only according to the amount of a quantity but also according to the direction in which the quantity is measured.
- Because a straight line is always the shortest distance between two places, the displacement vector is always a straight line from the beginning point to the current position of the item.
Conclusion
When it comes to position vectors, they are described as vectors that signify either the position or the location of any given point in relation to any arbitrary reference point such as the origin. The direction of a position vector is always pointing away from the origin of the vector and towards a specific location. The displacement vector of an object is the vector that represents the change in the item’s position vector. Allowing for the possibility that an object is at location A at time = 0 and at point B at time = t. Because a straight line is always the shortest distance between two places, the displacement vector is always a straight line from the beginning point to the current position of the item.
