Question

Prove that the midpoint of the line segment connecting the points $\left(x_1,x_2,\dots ,x_n\right)$ and $\left(y_1,y_2,\dots ,y_n\right)$ in${\mathrm{ℝ}}^n$ is$\left(\frac{x_1+y_1}{2},\frac{x_2+y_2}{2},\dots ,\frac{x_n+y_n}{2}\right)$.

Short answer

As a result, the coordinates of the line segment's midpoint $L$is

$\left(\frac{x_1+y_1}{2},\frac{x_2+y_2}{2},\cdots ,\frac{x_n+y_n}{2}\right)$

Step-by-step solution

Introduction

Consider the following points in$n$-dimensional plane:

$$\begin{gathered} P\left(x_1,x_2,\dots ,x_n\right) \\ Q\left(y_1,y_2,\dots ,y_n\right) \end{gathered}$$

Consider a line segment $L$joining the points $P$ and $Q$.

The objective is to prove that the midpoint of the line segment $L$ is,

$$

\left(\frac{x_{1}+y_{1}}{2}, \frac{x_{2}+y_{2}}{2}, \ldots, \frac{x_{n}+y_{n}}{2}\right)

$$