Position Vectors

Position vectors are the ones which tell us about the position of any point in space with respect to another point. Now, What do we need to know to specify someone's position?

Well Lets take an example: Suppose a thief steals a treasure and hides it then gets caught by the police. Police interrogates him and now he tells the location of the treasure like this. " I hid he treasure 10 feet away in east direction from the old Church in Mumbai". If we analyse this sentence then it will tell us all the necessary ingredients needed to make a position vector.

At First we need a starting point (Old Church in Mumbai), we can call this as a reference point( Often taken as origin).

Then we need directions, were to go?? in this case it was East. When we talk about directions then there are infinite directions we can talk about. but here we can use a fact that our space is a three dimensional space. That means we can at maximum has three mutually exclusive and independent directions originating from a point and all the directions can be broken in terms of these three directions.

Lastly we need to know the distance moved.

One of the simplest things we can do with a vector is encode a displacement. For instance, if we undertake the displacement 1 unit right, 2 units down, 3 units forward, we can call that action \(\vec{m}_1\). We can use it to move away from any position we might currently be located in.

Regardless of where we are, we will end up \(\sqrt{1^2+2^2+3^2}\) units away, along the direction that is 1 unit to the right, 2 units down, and 3 units forward from our current position.

It is important to realize that these displacements vectors are not positions, they are displacements. However, if we agree on a common point of reference, then we can encode position using a vector and a point.

For example, if we start at \(\langle 0, 0, 0 \rangle\), and we undertake the displacement \(\vec{m_1}\), then we move from the origin to the point \(\langle 1, 2, 3 \rangle\) (using the Cartesian coordinate system).

Therefore, when we specify an origin, and a set of displacements, we can keep track of the position of objects in space.