Vector, Displacement, Distance

Introduction to vector algebra

Vector is a powerful mathematical tool that helps us to describe motion. For simplicity, a vector can be thought as a straight arrow. It has a length (or magnitude) and a direction. The starting point of a vector does not matter in vector algebra. You can parallelly move a vector and consider it as the same vector. This motion, in which all points of a body move uniformly in the same direction, is called translational motion.

Addition and subtraction of two vectors are a bit more interesting than those of two numbers. Imagine the addition of a vector $\vec{a}$ and a vector $\vec{b}$ as a two-step guide in a navigation app. First walk the length of $\vec{a}$ along the direction of $\vec{a}$ , then walk the length of $\vec{b}$ along the direction of $\vec{b}$: a vector that points from the original start point to the final destination can be called $\vec{a}+\vec{b}$.

Only when $\vec{a}$ and $\vec{b}$ share a direction, $\vec{a}+\vec{b}$ has a magnitude equal to the sum of the magnitudes of $\vec{a}$ and $\vec{b}$. The subtraction of two vectors are very similar, after flipping the direction of the subtrahend vector.

Displacement and distance

For a path of motion, displacement is the vector that points from the original start point to the final destination, while distance is only a number that describes the length of the path.

In a special case, such as going grocery shopping, you leave your place and eventually come back. If we consider this part of history, you have travelled quite a few miles, but your displacement is simply a zero vector $\vec{0}$. Think about it: does a navigation app usually provide distance or displacement?