Position And Displacement Vectors

One of the primary things to start with Kinematics is to know what position vector is, what displacement vector is, and what the difference between position vector and displacement vector is.

Position Vector:

Position vector is used to specify the position of a certain body. Knowing the position of a body is vital when it comes to describe the motion of that body. The position vector of an object is measured from the origin, in general. Suppose an object is placed in the space as shown:

Position Vector

Position vector \((\vec{r})=x\hat{i}+y\hat{j}+z\hat{k}\)

Where,

\(\hat{i}\)= unit vector along x direction

\(\hat{j}\)= unit vector along y direction

\(\hat{k}\)= unit vector along z direction

So if an object is at a certain point P (say) at a certain time, its position vector is given as described above.

Displacement Vector:

The change in the position vector of an object is known as displacement vector. Suppose an object is at point A at time = 0 and at point B at time = t. The position vectors of the object at point A and at point B are given as:

Position vector at point \(A=\hat{r_{A}}=5\hat{i}+3\hat{j}+4\hat{k}\)

Position vector at point \(B=\hat{r_{B}}=2\hat{i}+2\hat{j}+1\hat{k}\)

Now, the displacement vector of the object from time interval 0 to t will be:

\(\hat{r_{B}}-\hat{r_{B}}=-3\hat{i}-\hat{j}-3\hat{k}\)

The displacement of an object can also be defined as the vector distance between the initial point and the final point. Suppose an object travels from point A to point B in the path shown in the black curve:

Displacement Vector

The displacement of the particle would be the vector line AB, headed in the direction A to B. The direction of displacement vector is always headed from initial point to the final point.

Fun fact: Displacement can also be said as the shortest distance between two points.

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Practise This Question

An ant moves from A to B then to c, what is the distance (d) and displacement (s) of the ant?