How to check if a point belongs to a segment in $\mathbb{R}^3$?

Question

I have a very simple question. Consider the segment $x$ given by $\{(\frac{1}{3},\frac{1}{3},\frac{1}{3}),(1,0,0)\}$. Then, my question is: does the point $(\frac{2}{3},\frac{1}{3},0)$ belong to such a segment $x$? Regardless of what the answer is (either yes or no), can you please show how to solve this question in general terms (as framed in the title)? Thank you all in advanced.

Answer 1

HINT

Indicating with $P=(\frac{1}{3},\frac{1}{3},\frac{1}{3}) \quad Q=(1,0,0)$

• write the parametric equation for the segment $P+t(Q-P)$
• equate to the coordinates $(\frac{2}{3},\frac{1}{3},0)$ and verify whether a solution exists for $t\in[0,1]$