Position and Displacement Vectors

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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 changes.

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:

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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:

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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 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:

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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:

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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. 

FAQ (Frequently Asked Questions)

Question 1: Specify the Direction of the Vector.

Answer: Whenever we talk about vectors, actually it is all about directions.

We know that the direction of a position vector is always initiated from origin to the point. So, if the origin is O, then the position vector of A, OA (Here, OA is the position of point A with respect to the origin). It is given by:

= x₁ (i)+ y₂ (j) and that for B is x₂ (i) - y₁(j).

Question 2: What Type of Quantity of Displacement and Velocity is?

Answer: Displacement and velocity in two or three dimensions are elementary extensions of the one-dimensional physical quantities. However, they are vector quantities, so for calculations with them, we have to follow the rules of vector algebra.

Question 3: What is the Distance Between Two Position Vectors in the Sky?

Answer: Let’s consider the radius of Earth as 6370 km, and the length of each position vector of the satellite from the Earth is 6770 km. We draw the two position vectors from the center of Earth, which is the origin with the X-axis as East and the Y-axis as North.

Question 4: Define a Vector.

Answer: A vector is represented by an arrow (head) whose direction is the same as that of the quantity and its length (tail) is proportional to the quantity’s magnitude.