Position vs Direction Vector

Every vector has a position vector, even vectors at the origin. We omit the origin for vectors because it’s easier

Visually, a vector is a line segment with a tip and a tail in the 2D plane. The vector is a straight line, so if I give you the coordinate of the tip of the vector and the coordinate of the tail, you only have to draw a straight line between those two points to draw our vector or line segment.

If the vector is at the origin, the tail is at $(0,0)$. There is no need to specify that each time, thus we only need the coordinate of the tip.

Physically, we can move in the same direction by walking in parallel lines or by walking in the same line

If two racing cars A and B are moving east during a race in straight lines, intuitively, we say that the cars are moving in the same direction.

Consider a military march where soldiers are moving in the same line forward. We consider them to be walking in the same direction too. While the soldiers are walking on the same line, and the cars speed on parallel but different lines, we say that they are both going in the same direction. How do we define direction so that it encompasses those two scenarios?

The answer is the direction vector. It’s a vector centered at the origin of the plane that specifies the direction in which vectors are going. If a vector is at the origin, we consider it to also be its own direction vector.

What should we do when the vector is not at the origin? Remember from our intuition, we want both parallel vectors to have the same direction. As such, we must define the direction vector in a way that makes two parallel vectors codirectional.

To find the direction vector, we take the position vector and translate it toward the origin. You keep the size and orientation of the vector the same. The translated vector at the origin is the direction vector.

Now if two vectors have the same orientation but aren’t collinear like our two cars earlier, their direction vectors will point in the same direction.

Algebraically, if $\vec{u}$ and $\vec{v}$ are position vectors, the direction vector from $\vec{v}$ to $\vec{u}$ is: $\vec{u} - \vec{v}$

Convince yourself using vector addition by the triangle or parallelogram method that $\vec{u} - \vec{v}$ is indeed the direction vector we described visually earlier.

We use the unit direction vector $\hat{v} = \frac{\vec{v}}{|\vec{v}|}$ to specify direction precisely because it gives us pure direction without magnitude information, allowing us to separate “which way” from “how far.”