Position vs Direction Vectors

We often normalize direction vectors to create unit direction vectors, which have a magnitude of exactly 1. The unit direction vector is calculated as $\hat{v} = \frac{\vec{v}}{|\vec{v}|}$, where $|\vec{v}|$ represents the magnitude of vector $\vec{v}$.

Unit direction vectors are particularly useful because they specify pure direction without any magnitude information. This allows us to separate the concepts of “which way” and “how far,” making calculations cleaner and more intuitive. When we need a vector in a specific direction with a particular magnitude, we can simply multiply the unit direction vector by the desired scalar value.

Position Vectors

Every vector can be represented as a position vector, which starts from the origin $(0,0)$ and points to a specific coordinate. This representation is convenient because instead of specifying both the tail and tip coordinates of a vector, we only need to specify where the tip is located. When we say a vector has components $(3,4)$, we mean it starts at the origin and ends at the point $(3,4)$.

For example, if we have a vector from point $A(1,2)$ to point $B(4,6)$, we can represent this as the position vector $\vec{v} = (3,4)$. This vector captures the same displacement but is standardized to start from the origin.

Understanding Direction

Consider two scenarios where we intuitively say objects are moving in the “same direction.” First, imagine racing cars A and B both moving east, but on parallel lanes. Second, picture soldiers marching forward in a single-file column. Even though the cars travel on different parallel lines while the soldiers travel on the same line, we naturally say both groups are moving in the same direction.

This intuition reveals an important concept: direction should be independent of the specific path or starting position. Two vectors should be considered to have the same direction if they are parallel, regardless of where they are positioned in the plane.

Direction Vectors

A direction vector captures only the orientation of movement, regardless of starting position. When we want to find the direction vector of any displacement, we calculate it using vector subtraction. If we have two position vectors $\vec{u}$ and $\vec{v}$, the direction vector from $\vec{v}$ to $\vec{u}$ is simply $\vec{u} - \vec{v}$.

This makes sense when we think about vector addition. If we start at point $\vec{v}$ and want to reach point $\vec{u}$, we need to add some displacement vector to $\vec{v}$. That displacement vector is exactly $\vec{u} - \vec{v}$, since $\vec{v} + (\vec{u} - \vec{v}) = \vec{u}$.

The key insight is that parallel vectors will always have the same direction vector (or scalar multiples of the same direction vector), which matches our intuitive understanding that parallel movements represent the same direction.

Unit Direction Vectors

We often normalize direction vectors to create unit direction vectors, which have a magnitude of exactly 1. The unit direction vector is calculated as $\frac{\vec{v}}{|\vec{v}|}$, where $|\vec{v}|$ represents the magnitude of vector v.

Unit direction vectors are particularly useful because they specify pure direction without any magnitude information. This allows us to separate the concepts of “which way” and “how far,” making calculations cleaner and more intuitive. When we need a vector in a specific direction with a particular magnitude, we can simply multiply the unit direction vector by the desired scalar value.