## Task Condition

Are the vectors $a, b$ and $c$ coplanar?

$a=\{7 ; 3 ; 4\}$

$b=\{-1 ;-2 ;-1\}$

$c=\{4 ; 2 ; 4\}$
## Solution

For three vectors to be coplanar (lie in the same plane or parallel planes), it is necessary and sufficient that their scalar triple product $(a, b, c)$ be equal to zero.

$$
\begin{aligned}
& (a, b, c)=\left|\begin{array}{ccc}
7 & 3 & 4 \\
-1 & -2 & -1 \\
4 & 2 & 4
\end{array}\right|= \\
& =7 \cdot\left|\begin{array}{cc}
-2 & -1 \\
2 & 4
\end{array}\right|-3 \cdot\left|\begin{array}{cc}
-1 & -1 \\
4 & 4
\end{array}\right|+4 \cdot\left|\begin{array}{cc}
-1 & -2 \\
4 & 2
\end{array}\right|= \\
& =7 \cdot(-6)-3 \cdot 0+4 \cdot 6=-42+24=-18
\end{aligned}
$$

Since $(a, b, c)=-18 \neq 0$, the vectors $a, b$ and $c$ are not coplanar.