0

I'm studying vectors in Analytic Geometry and we were given an exercise to verify if three points are collinear. Once two vectors $\vec v$ and $\vec u$ are collinear if $\exists k \in \mathbb{R}: \vec v=k\vec u$, two vectors are collinear if they are parallel.

Given the points $P(3, 2, 1)$, $Q(-2, 5, 2)$ and $S(8, -1, 0)$,

$\overrightarrow {PQ} = Q - P = (-2, 5, 2)-(3, 2, 1)=(-5, 3, 1)$

$\overrightarrow {QS} = S -Q = (8, -1, 0)-(-2, 5, 2)=(10, -6, -2)$

Once $-2 \overrightarrow {PQ}=\overrightarrow {QS}$, we have $\overrightarrow {QS}\parallel\overrightarrow {PQ}$ and hence $P, Q$ and $S$ are collinear.

In analytic geometry collinear vectors and parallel vectors seems to be the same thing (Note: I don't know about linear algebra), but is there a difference between them? I know that there is an answer here Difference between collinear vectors and parallel vectors? , but I would like other ones.

欲しい未来
  • 891
  • It really depends if you care about where the tail of the vectors are. If two vectors have the same tail location then being parallel $\iff$ being colinear. The diagram in the answer to the question you linked sums it up nicely. – Graviton Sep 25 '20 at 02:35
  • 1
    I always thought that "collinear" means on the same line. So if two parallel vectors have a common point(s), they are collinear. – Vasili Sep 25 '20 at 02:44

0 Answers0