Use vector addition, subtraction, and scalar multiplication to show that the midpoint between the two points $\mathbf{x}$ and $\mathbf{y}$ is $\frac{\mathbf{x} + \mathbf{y}}{2}$.

Solution

Let's call the midpoint $\bfz$. Our goal is to relate $\bfz$ to $\bfx$ and $\bfy$.

We choose to represent all line segments in this problem as vectors.

Now, we interpret the fact that $\bfz$ is a midpoint in terms of vectors.

Similarly, the vector going from $\bfx$ to $\bfz$ is $\bfz - \bfx$.

Because $\bfz$ is the midpoint of $\bfx\bfy$, the vector going from $\bfx$ to $\bfz$ has the same length and direction as the vector going from $\bfz$ to $\bfy$.

Hence, we can write that $$ \bfz - \bfx = \bfy - \bfz.$$

Visually, this equality can be depicted as:

Adding $\bfz$ and $\bfx$ to both sides, we get \begin{align} 2\bfz &= \bfx + \bfy. \end{align}

After dividing by 2, we conclude $$ \bfz = \frac{\bfx + \bfy }{2}.$$