## Problem Statement

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

$a=\{-1 ; 4 ; 2\}$

$b=\{3 ;-2 ; 6\}$

$c_{1}=2 a-b$

$c_{2}=3 b-6 a$
## 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 $c_{2}=-3(2 a-b)=-3 c_{1}$ for any $a$ and $b$.

That is,

$c_{1}=-\frac{1}{3} \cdot c_{2}$

, which means the vectors $c_{1}$ and $c_{2}$ are collinear.

## Problem Kuznetsov Analytical Geometry 3-10