## Task Condition

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

$x=\{-2 ; 4 ; 7\}$

$p=\{0 ; 1 ; 2\}$

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

$r=\{-1 ; 2 ; 4\}$
## 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:

$$
\left\{\begin{array}{l}
\beta-\gamma=-2 \\
\alpha+2 \gamma=4 \\
2 \alpha+\beta+4 \gamma=7
\end{array}\right.
$$

Add the second row multiplied by -2 to the third row:

$$
\begin{aligned}
& \left\{\begin{array}{l}
\beta-\gamma=-2 \\
\alpha+2 \gamma=4 \\
\beta=-1
\end{array}\right. \\
& \left\{\begin{array}{l}
-1-\gamma=-2 \\
\alpha+2 \gamma=4 \\
\beta=-1
\end{array}\right. \\
& \left\{\begin{array}{l}
\gamma=1 \\
\alpha+2 \gamma=4 \\
\beta=-1
\end{array}\right.
\end{aligned}
$$

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

The desired decomposition:

$x=2 p-q+r$

Problem Kuznetsov Analytical Geometry $2-1$