## problem statement

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

$$
\begin{aligned}
& a=\{-2 ;-3 ;-2\} \\
& b=\{1 ; 0 ; 5\} \\
& c_{1}=3 a+9 b \\
& c_{2}=-a-3 b
\end{aligned}
$$
## 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 $c_{1}=-3(-a-3 b)=-3 c_{2}$ for any $a$ and $b$.

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

## Problem Kuznetsov Analytical Geometry 3-9