## Task Condition

Write the decomposition of vector $x$ in terms of vectors $p, q, r$:

$x=\{-19 ;-1 ; 7\}$

$p=\{0 ; 1 ; 1\}$

$q=\{-2 ; 0 ; 1\}$

$r=\{3 ; 1 ; 0\}$
## Solution

The desired decomposition of vector $x$ is:

$x=\alpha \cdot p+\beta \cdot q+\gamma \cdot r$

Or in the form of a system:

$$
\left\{\begin{array}{l}
\alpha \cdot p_{1}+\beta \cdot q_{1}+\gamma \cdot r_{1}=x_{1} \\
\alpha \cdot p_{2}+\beta \cdot q_{2}+\gamma \cdot r_{2}=x_{2} \\
\alpha \cdot p_{3}+\beta \cdot q_{3}+\gamma \cdot r_{3}=x_{3}
\end{array}\right.
$$

We obtain:

$$
\left\{\begin{array}{l}
-2 \beta+3 \gamma=-19 \\
\alpha+\gamma=-1 \\
\alpha+\beta=7
\end{array}\right.
$$

Subtract the second row from the third:

$$
\left\{\begin{array}{l}
-2 \beta+3 \gamma=-19 \\
\alpha+\gamma=-1 \\
\beta-\gamma=8
\end{array}\right.
$$

Add the third row multiplied by 2 to the first row:

$$
\left\{\begin{array}{l}
\gamma=-3 \\
\alpha+\gamma=-1 \\
\beta-\gamma=8
\end{array}\right.
$$

$$
\begin{aligned}
& \left\{\begin{array}{l}
\gamma=-3 \\
\alpha-3=-1 \\
\beta-(-3)=8
\end{array}\right. \\
& \left\{\begin{array}{l}
\gamma=-3 \\
\alpha=2 \\
\beta=5
\end{array}\right.
\end{aligned}
$$

The desired decomposition:

$x=2 p+5 q-3 r$

## Problem Kuznetsov Analytic Geometry 2-7