## Problem Statement

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

$$
\begin{aligned}
& a=\{5 ; 0 ;-1\} \\
& b=\{7 ; 2 ; 3\} \\
& c_{1}=2 a-b \\
& c_{2}=3 b-6 a
\end{aligned}
$$
## 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 $c_{2}=-3(2 a-b)=-3 c_{1}$ 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 3-11