Position and Displacement Vectors

Introduction

If the position of any object changes relative to the time it is said to be in motion. Motion of any object that happens in three ways like one-dimensional motion, two-dimensional motion, and three-dimensional motion. Movement of a train on a straight track, circular motion of the top are some examples for motion. This motion of particles is best described by the law of mechanics. Three laws were proposed for the understanding of the movement of an object. When there is a motion then there will be a displacement. In this tutorial, we are going to discuss the topic of vectors, position vector and displacement vectors, the differences and similarities between the position and displacement vectors, properties of the two vectors in detail. Starting with the vectors.

What is a vector?

The physical quantities are classified as scalar and vector. Scalar is a physical quantity which describes only the magnitude of the quantity. Some of the examples of scalar quantity are length, mass, speed etc… Vector describes both the magnitude and direction of the quantities. Some of the examples of vector quantities are displacement, weight, velocity etc.. The direction of the movement of the object is denoted by an arrow above the quantity. Generally, vectors have two components such as horizontal component and vertical component. Some types of vectors are equal vectors, zero or null vector, negative vector, unit vector, coplanar vector, collinear vector, position vector and displacement vector.

What is the position and displacement vector? Explain with example

Position vector

The change in position denotes the motion. The rectangular coordinates system is used to define the coordinates of the position of the object. The position of the object with respect to other objects is located by the position vector. This position is located by taking origin as an initial position. The position vector is denoted by a straight line which is connected by the fixed object and the object which is in motion. The magnitude or direction or both of the position vector varies due to the motion of the object. Let us consider a point P in a three-dimensional rectangular coordinate system whose positions are denoted as P(x_1,y_1,z_1). The coordinates of the origin are (0,0,0). The position vector that is drawn from the origin to point P is OP and is denoted as x_1 i+y_1 j+z_1 k. Here i,j, and k are the unit vectors along X,Y, and Z axes.

Displacem ent vector

If the particle moves from one location to another location then the particle is said to be in motion. The distance traveled in a certain time following the direction of motion is called displacement. Thus the displacement is a vector quantity. An arrow mark is used to express the direction of displacement. As it is a vector quantity its direction is denoted with the help of arrow marks. The displacement vector defines the displacement of the object with respect to one another.

What is the difference between position and displacement vector?

• Both the position vector and the displacement vector denote a vector quantity.

• In this the position vector is used only to spotify the position of the object with respect to the reference point located at the origin.

• The displacement vector gives the change in position of the object. The position vector is fixed but the displacement vector is a varying vector.

How to find positions and displacement vectors?

To find position vector

The coordinates of the object are necessary for the calculation of position vectors. Let the position vector of A is (x_1,y_1,z_1) for B is (x_2,y_2,z_2). The position vector of the object at point A is OA. Similarly the position vector of B with respect to origin is OB. The position vector OA and OB are noted as x_1 i+y_1 j+z_1 k and x_2 i+y_2 j+z_2 k. The position vector of A with respect ot to B is given by,

$$\mathrm{AB=OB-OA }$$

$$\mathrm{AB=(x_2-x_1)i+(y_2-Y_1)j+(z_2-z_1).}$$

To find displacement vector

The vector distance of the object from the initial point to the final point is given by the displacement vectors. The displacement vector is used to understand the change in position of the position vectors. Let us take the position vectors of A and B are 4i+2j-3k and 2i-j+5k. The displacement vector is calculated by the following as,

$$\mathrm{AB = (4-2)i+(2-1)j+(3-5)k}$$

$$\mathrm{AB=2i+j-2k}$$

Displacement vector

In this AB denotes the displacement vector of the object.

Conclusion

In this tutorial, we learned about vectors, Position and Displacement vectors, their properties and differences. Scalar is a quantity which is denoted only by the magnitude. But vector quantity denotes both the magnitude and direction. Position vector gives the location of the object. When the position of the object changes then there is a displacement. The difference between the two position vectors of the object gives the displacement vector. Finding procedure for the position vector and the displacement vector were also discussed.

FAQs

1. What is a path length?

The distance between the final initial and final position of the object denotes the path length. In the following figure let us consider the motion of the cycle from A to B. The path length of the cycle to cover the distance from A to B is 4m. Let us consider that the cycle moves from A to B then B to C and then from C to A. The total distance covered or the path length the cycle covered is 4 + 3 + 5 = 12m. If the object moves in a circular path then the total distance covered by the object is given by 2πr.

2. Can vectors be negative?

Vectors can be negative if the direction of the vector is in the opposite direction to the reference direction.

3. What happens to the angular motion when the force is stopped suddenly?

If the object is moving in a circular path, it is mainly due to the centripetal force on it. That is why it is rotating in the same path. If the force is stopped then the object undergoes linear motion in the tangential direction.

4. Why does the position vector come under a fixed vector?

If the initial and the tail of the vectors are fixed then it is called a fixed vector. As the initial point of the position vector fixed at origin it may be a fixed or localised vector.

5. Why is the displacement a vector quantity while distance is a scalar?

Distance is the measurement taken by considering only the magnitude of the length of the path whereas the displacement is a measurement in which both magnitude and direction are taken into consideration.