Co-planar vectors

Vectors that are parallel to one plane. A necessary and sufficient condition for the co-planarity of three vectors

$$ \mathbf a _ {1} = \{ x _ {1} , y _ {1} , z _ {1} \} ,\ \ \mathbf a _ {2} = \{ x _ {2} , y _ {2} , z _ {2} \} ,\ \ \mathbf a _ {3} = \{ x _ {3} , y _ {3} , z _ {3} \} $$

is the equation

$$ \left | \begin{array}{lll} x _ {1} &y _ {1} &z _ {1} \\ x _ {2} &y _ {2} &z _ {2} \\ x _ {3} &y _ {3} &z _ {3} \\ \end{array} \right | = 0. $$