## Problem Statement

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

$a=\{-1 ; 2 ; 8\}$

$b=\{3 ; 7 ;-1\}$

$c_{1}=4 a-3 b$

$c_{2}=9 b-12 a$
## 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 $c_{2}=-3(4 a-3 b)=-3 c_{1}$ for any $a_{\text {and }} b$.

That is,

$$
c_{1}=-\frac{1}{3} \cdot c_{2}, \text { which means the vectors } c_{1 \text { and }} c_{2-} \text { are collinear. }
$$

Problem Kuznetsov Analytical Geometry 3-31