## Problem Statement

Are the vectors $c_{1 \text { and }} c_{2}$, constructed from vectors $a \text{ and } b$, collinear?

$a=\{8 ; 3 ;-1\}$

$b=\{4 ; 1 ; 3\}$

$c_{1}=2 a-b$

$c_{2}=2 b-4 a$
## Solution

Vectors are collinear if there exists a number $\gamma$ such that $c_{1}=\gamma \cdot c_{2}$. That is, vectors are collinear if their coordinates are proportional.

It is not hard to notice that $c_{2}=-2(2 a-b)=-2 c_{1}$ for any $a$ and $b$.

That is, $c_{1}=-2 \cdot c_{2}$, which means the vectors $c_{1}$ and $c_{2}$ are collinear.

## Problem Kuznetsov Analytical Geometry 3-18