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Position and Displacement Vectors - A Complete Guide

Position and Displacement Vectors - A Complete Guide

Vishal kumarUpdated on 02 Jul 2025, 05:01 PM IST

A vector may be an extent that has both magnitude and direction. Vectors allow us to explain the quantities which have both direction and magnitude. For instance velocity, and position. In this article we will discuss about, what is Position? Or definition of position. What is position vector? Define position vector with its formula. How to find Position vector? How can we describe the position of an object? What is displacement vector? Give its formula. We define the position vector as a straight-line possessing one end fixed to an object and therefore the other end attached to a moving point (marked by an arrowhead) and wont to represent the position of the purpose relative to the given object. Since, the point moves, the position vector switches long or in direction, and sometimes both length and direction changes. Are you know, what is Position??? So Now We define Position (Edge) is that the location of the thing (whether it is a person, a ball, or a particle) at a given moment in time. Vector physics vallah is anything which also has direction. A displacement vector is an idea from vectors. It’s a vector. It indicates the direction and distance traveled with a line.

Position and Displacement Vectors - A Complete Guide
Position and Displacement Vectors

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What is Position Vector?

A position or edge(Position) vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually signified as x, r, or s, it point to the straight line segment from O to P.

Radius vector r represent the postion of point P (x, y, z) with repect to origin O

In the above statement, we took a frame of reference to represent your journey from the origin, i.e., your home to succeed in your favorite destinations, first, Tamilnadu, then, Bengluru.
Each destination is marked by an arrow on the graph, which changes or varies as you modify your destination, below represent the same:


Hence, your position vector changes, i.e., twice or twice the length,
So, along the X-axis, the position vector is: ‘i (cap)’ and along the Y-axis, it's ‘j (cap)’. Since the position resultant is represented by r→
, therefore the resultant of the position vectors along with coordinate axes are going to be as follows:

r→ + I (cap) + j (cap) ……………………… (1)

Now In these Articles, the points we have to discuss about Position Vector are:

  • Explain what's an edge (Position) Vector?

  • And how to find out the Position Vector?

  • How can we describe the position of an object?

Often, we've noticed that the vectors start at the origin and terminate at any arbitrary point are referred to as position vectors. These are said to work out the position of some extent with regard to the origin of that time. The orientation/direction of the vectors or the position vector generally points from the origin towards the given point. Within the frame of reference of c\Cartesian if point O is that the origin and Q is a few points that's x1, y1, then the directed position vector from point O to point Q is identified as OQ. Within the space which is three-dimensional if O = (0, 0, and 0) and Q = (x1, y1, z1), then the position vector denoted by r of point Q is represented is:

r = x1i + y1j + z1k

Now let’s suppose that we've two vectors, that are A and B, with position vectors we write a = (2, 4) and b = (3, 5) respectively. We will then write the coordinates of both the vectors that are A and B as:

A = (2, 4), B = (3, 5)

Here before determining the position vector of some extent we first got to determine the coordinates of these particular points. Let’s suppose that we've two points, namely M and N. where the purpose M = (x1, y1) and N = (x2, y2). Next we would like to seek out here the position vector that too from point M to point N the vector MN. To work out this position vector, we simply subtract the corresponding components of M from N:

Written as: MN = (x2-x1, y2-y1)

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Position Vector Formula:

If we consider some extent denoted by letter P. Which has the coordinates that are xk, yk within the xy-plane and another point written as Q. Which has the coordinates denoted by xk+1, yk+1.

The formula which is to work out the position vector that's from P to Q is written as:

PQ = ((xk+1)-xk, (yk+1)-yk)

We can now remember the position vector that's PQ which generally refers to a vector that starts at the purpose P and ends at the purpose Q. Similarly if we would like to findout the position vector that's from the purpose Q to the purpose P then we will write:

QP = (xk – (xk+1), yk – (yk+1))

How can we describe the position of an object?

The changes in position of an object with reference to its surroundings during a given interval of your time. An object is moving if its position relative to a hard and fast point is changing. Even things that indicate to be at rest move.

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Commonly Asked Questions

Q: How does the concept of reference frame relate to position vectors?
A:
A reference frame is crucial for position vectors as it provides the origin and coordinate system for measuring position. The same object can have different position vectors in different reference frames, highlighting the importance of specifying the frame when discussing position.
Q: How does dimensional analysis apply to position and displacement vectors?
A:
Both position and displacement vectors have the dimension of length [L]. This is important for checking the consistency of equations and ensuring that physical quantities are correctly related in formulas involving these vectors.
Q: How do position and displacement vectors behave in circular motion?
A:
In circular motion, the position vector continuously changes as the object moves around the circle. However, the displacement vector can vary depending on the time interval considered. For a complete revolution, the displacement is zero, despite the changing position.
Q: How do position and displacement vectors relate to coordinate systems?
A:
Position and displacement vectors are typically expressed in terms of coordinate systems (e.g., Cartesian, polar). The choice of coordinate system can simplify calculations and affect how the vectors are represented, but doesn't change their physical meaning.
Q: Can an object have a changing position vector but constant displacement vector?
A:
No, if an object's position vector is changing, its displacement vector must also change. Displacement represents the change in position, so any alteration in position necessarily affects the displacement vector.

Displacement Vector

The change within the position vector of an object is understood because the displacement vector. Let us consider an object is at point A at time = 0 and at point B at time = t. The position vectors of the thing at point A and at point B are given as:

Position vector at point A = r^A = 5 i^ + 3 j^+ 4 k^
Position vector at point B = r^B = 2 i^ + 2 j^+ 1 k^
Now, the displacement vector of the thing from interval 0 to t will be:
r^A - r^B = - 3 i^ - j^+ 3k^
The displacement of an object also can be defined because the vector distance between the initial point and therefore the final point. Suppose an object travels from point A to point B within the path shown within the black curve:


The displacement of the particle would be the vector line AB, lead inside the direction A to B. The direction of the displacement vector is usually headed from the initial point to the ultimate point.

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NCERT Physics Notes:

Commonly Asked Questions

Q: What is the difference between position and displacement vectors?
A:
Position vectors indicate an object's location relative to a fixed origin, while displacement vectors represent the change in position between two points. Position vectors give absolute location, while displacement vectors show relative movement.
Q: Why is displacement considered a vector quantity?
A:
Displacement is a vector quantity because it has both magnitude and direction. It represents not just how far an object has moved, but also in which direction, making it fundamentally different from scalar quantities like distance.
Q: Can displacement be greater than the total distance traveled?
A:
No, displacement cannot be greater than the total distance traveled. Displacement is the straight-line distance between start and end points, while total distance includes all path lengths. The displacement is always less than or equal to the total distance.
Q: How do you add displacement vectors?
A:
Displacement vectors are added using vector addition, typically through the tip-to-tail method or component method. This involves considering both magnitude and direction, not simply adding scalar values.
Q: Can displacement be zero even if an object has moved?
A:
Yes, displacement can be zero even if an object has moved. This occurs when the object returns to its starting point. The total distance traveled may be non-zero, but the net displacement (straight-line distance between start and end points) is zero.

Frequently Asked Questions (FAQs)

Q: What's the significance of displacement in understanding conservation laws?
A:
Displacement plays a crucial role in many conservation laws. For example, in conservative force fields, the work done (which involves displacement) is independent of the path taken, leading to the conservation of mechanical energy. This highlights the fundamental nature of displacement in physics.
Q: Can displacement be negative?
A:
The magnitude of displacement is always positive, but the displacement vector can be negative in the sense that it can point in the negative direction of a coordinate axis. This emphasizes the importance of considering both magnitude and direction in vector quantities.
Q: What's the significance of initial and final position vectors in calculating displacement?
A:
The initial and final position vectors are crucial for calculating displacement. Displacement is found by subtracting the initial position vector from the final position vector. This subtraction accounts for both the magnitude and direction of the movement.
Q: How do position and displacement vectors behave in two-dimensional motion?
A:
In two-dimensional motion, position and displacement vectors have components in two perpendicular directions. The total displacement is the vector sum of these components, often calculated using the Pythagorean theorem and trigonometry.
Q: How do position and displacement vectors behave in simple pendulum motion?
A:
In simple pendulum motion, the position vector traces an arc. The displacement vector changes continuously in both magnitude and direction. At the extremes of the swing, the displacement is maximum but momentarily zero velocity.
Q: What's the significance of displacement in defining mechanical waves?
A:
In mechanical waves, displacement refers to the distance and direction that a point on the medium moves from its equilibrium position. This concept is fundamental to understanding wave propagation and behavior.
Q: How do position and displacement vectors relate to the concept of center of mass?
A:
The center of mass of a system can be represented by a position vector. The displacement of the center of mass is often used to simplify the analysis of complex systems, treating them as single particles in many calculations.
Q: Can displacement be used to determine the direction of motion?
A:
Displacement can indicate the overall direction of motion between start and end points, but it doesn't provide information about the path taken or any changes in direction during the motion. It gives the net effect of the motion, not its full history.
Q: How do position and displacement vectors behave in damped oscillations?
A:
In damped oscillations, both position and displacement vectors behave similarly to simple harmonic motion, but with decreasing amplitude over time. The displacement vector's magnitude gradually decreases as energy is dissipated from the system.
Q: How do position and displacement vectors relate to the concept of torque?
A:
Torque is calculated using the position vector from the axis of rotation to the point where force is applied. While displacement isn't directly used in the torque formula, understanding both position and displacement is crucial for analyzing rotational motion.
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