## Problem Statement

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

$a=\{5 ;-1 ;-2\}$

$b=\{6 ; 0 ; 7\}$

$c_{1}=3 a-2 b$

$c_{2}=4 b-6 a$
## 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 $c_{2}=-2(3 a-2 b)=-2 c_{1}$ for any $a$ and $b$.

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

Problem Kuznetsov Analytical Geometry 3-23