Position Vector Definition
We all deal with a graph, mark a line from the origin and reach the other end till out requirement. All these requirements are done on the coordinate system. So, the coordinate where our line indicated by an arrow terminates is the coordinate of this ray.
Let’s consider, you started your journey from home to reach your favorite destination and then route to another destination, so your arrow is changing both of its length and direction, which means your position vector is changing and in case, you choose the shortest path, i.e., displacement, you represent it by the displacement vector.
Position Vector Definition
We define the position vector as a straight-line having one end fixed to an object and the other end attached to a moving point (marked by an arrowhead) and used to represent the position of the point relative to the given object. As the point moves, the position vector changes in length or in direction, and sometimes both length and direction change.
An Introduction to Position Vector and Displacement Vector
In the study of our physical world, the concepts of position and displacement are the foundational topics for the chapter of motion. The concept of 'or 'position vector' has been adopted from Euclidean spaces or geometry and is also known as location vector or radius vector. The position of any point in space is expressed in terms of three coordinates, namely 'x,' 'y,' and 'z' distances from any arbitrary point denoted as 'O' or origin. The straight line from origin to the point is denoted by 'r' or 's'. In Physics this vector is used while describing an object in rest or in motion in space with reference to another object. depending upon the various locations at different instants of time the vector changes in length and direction accordingly.
The three coordinates described in vectors of each direction are also referred to as three dimensions. Displacement is any change in any of these vectors. In common language, we know displacement as any movement of an object from one place to another place by following a straight path. If any object doesn't follow a straight line path then the total path covered is measured as distance. And the displacement in this case would be the straight distance between the starting point and finishing point. While describing a displacement, which informs the shortest distance between two points, it is also important to mention the direction of the displacement to know the exact location of the final point. Thus when we denote the direction of a point then it is known as position vector and when the direction of a displacement is mentioned then it is known as displacement vector.
What is Position Vector?
In the above statement, we took a coordinate system to represent your journey from the origin, i.e., your home to reach your favorite destinations, first, Darjeeling, then, Karnataka.
Each destination is marked by an arrow on the graph, which changes or varies as you change your destination, below is the graph to represent the same:
(Image will be Uploaded Soon)
Hence, your position vector changes, i.e, two times or twice the length, and the direction of the position vector changes according to this scenario.
So, along the X-axis, the position vector is: ‘i (cap)’ and along the Y-axis, it is ‘j (cap)’. Since the position vector sum is represented by r\[^{\rightarrow }\], so the vector sum of the position vectors along the coordinate axes will be as follows:
r\[^{\rightarrow }\] = i (cap) + j (cap).....(1)
Displacement Vector Definition
A displacement vector is one of the important concepts of mathematics. It is a vector. It represents the direction and distance traveled by an object in a straight line. We often use the term ‘displacement vector’ in physics to showcase the speed, acceleration, and distance of an object traveling in a direction relative to a reference point or an object's starting position.
What is a Displacement Vector?
The displacement vector definition is very simple to understand. Let’s discuss the scenario, you decide to travel to two locations for office work in the minimum time possible, and both of these locations are adjacent to each other in the mid of two roads passing opposite each other. Now, you have to decide from which path you should go in order to reach in the required time, as there is a lot of traffic on the road and the thoughts of getting scolded by your boss. So, below is the visual schematic representation of your situation:
(Image will be Uploaded Soon)
So, here, the green line is the shortest path, which will help you reach the middle of the two roads and reach the two locations on time. So, a displacement vector represents the minimum distance to reach on time rather than taking a long path with a wastage of a large amount of time.
Now, after your work is done, you take an opposite oath, so here, your displacement isn’t changing, only the direction is. So, with the direction, the displacement vector changes in terms of direction, not in magnitude.
Displacement Vector
We know that the change in the position vector of an object is known as the displacement vector. Let’s suppose that an object is at the point P at time = 0 and at the point Q at time = t. The position vectors of the object at the point P and at point Q are represented in the following way:
Position vector at point P = r\[^{\rightarrow }\] P (cap) = 8i (cap) +5j (cap) + 3k (cap)....(a)
Position vector at point Q = r\[^{\rightarrow }\] Q (cap) = 2 (cap) +2j (cap) +1k (cap).....(b)
Now, the displacement vector of the object traveling from time interval 0 to t will be as follows:
r\[^{\rightarrow }\] Q (cap)−r r\[^{\rightarrow }\] P (cap) =− 6i (cap) − 3j (cap) −2k (cap)....(c)
Equation (c) is the displacement vector formula and the schematic representation of this equation is as follows:
(Image will be Uploaded Soon)
We can also define the displacement of an object as the vector distance between the initial point and the final/ultimate point of the destination. Suppose an object travels from point P to point Q in the path shown in the black curve:
(Image will be Uploaded Soon)
We can imagine that the displacement of the particle would be the vector line PQ, headed in the direction P to Q and the direction of the displacement vector is always initiated from the initial point and terminated to the final point.
The Final Words
One of the most important aspects of kinematics is the position vector and the displacement vector; also, the key differences between these two, about which we discussed in the above context.
The position vector specifies the position of a known body. Knowing the position of a body is paramount when it comes to describing its motion. However, the change or variation in the position vector is the displacement vector.
FAQs on Position and Displacement Vectors
1. What is a position vector in Physics?
A position vector is a vector that represents the location of a point in space relative to a fixed reference point, known as the origin. It is drawn as a straight line from the origin to the point. For a point P with coordinates (x, y, z), its position vector, denoted as r or OP, is given by r = xî + yĵ + zk̂, where î, ĵ, and k̂ are the unit vectors along the X, Y, and Z axes, respectively.
2. How is a displacement vector defined and how does it differ from a position vector?
A displacement vector represents the change in an object's position. It is the shortest straight-line distance from the initial point to the final point, indicating both magnitude and direction of the movement. While a position vector specifies a single location relative to the origin, a displacement vector describes the net result of a movement between two locations and is independent of the path taken.
3. How do you calculate the displacement vector between two points?
To calculate the displacement vector, you subtract the initial position vector from the final position vector. If an object moves from an initial position P₁ (with position vector r₁ = x₁î + y₁ĵ + z₁k̂) to a final position P₂ (with position vector r₂ = x₂î + y₂ĵ + z₂k̂), the displacement vector (Δr) is:
Δr = r₂ - r₁ = (x₂ - x₁)î + (y₂ - y₁)ĵ + (z₂ - z₁)k̂
4. Why is the origin crucial when defining a position vector but not for a displacement vector?
The origin is crucial for a position vector because its definition depends entirely on this reference point; changing the origin changes the position vector of a point. However, a displacement vector represents the difference between two position vectors. If the origin is shifted, both the initial and final position vectors change by the same amount, but their difference (the displacement vector) remains exactly the same. This makes displacement an independent quantity, describing only the movement itself.
5. Can an object have zero displacement but non-zero distance travelled? Explain with an example.
Yes, this is a key difference between the two concepts. Distance is a scalar quantity representing the total path length covered, while displacement is a vector representing the net change in position. A common example is an athlete running one full lap around a 400m circular track.
- Distance travelled: 400 metres.
- Displacement: Zero, because the athlete ends at the exact same point where they started, so their initial and final position vectors are identical.
6. What is the physical meaning of a negative displacement vector?
In physics, the negative sign on a vector simply indicates its direction. A negative displacement vector (e.g., -Δr) has the exact same magnitude as its positive counterpart (Δr) but points in the precise opposite direction. For example, if a displacement of +5m î means moving 5 metres east, a displacement of -5m î means moving 5 metres west.
7. Give a real-world example of how position and displacement vectors are used.
A prime example is GPS navigation. A GPS receiver calculates your position vector relative to a coordinate system fixed to the Earth (using signals from satellites). When you move from your home to your school, the GPS tracks the change. Your home has an initial position vector (r₁), and your school has a final position vector (r₂). The navigation system calculates the displacement vector (Δr = r₂ - r₁) to determine the most direct route and display your movement on a map.
8. If a particle's position vector changes in direction but its magnitude remains constant, what kind of path is it following?
If the magnitude of a particle's position vector (|r|) remains constant while its direction changes, the particle is moving along a path where its distance from the origin is always the same. In a two-dimensional plane, this path is a circle with the origin at its center. In three-dimensional space, it would be moving on the surface of a sphere centered at the origin. This is a fundamental concept in describing uniform circular motion.
