I am trying to write down the angle between two $1$-dimensional and $t$-dimensional subspaces of a normed space $L_p$. In particular, I am following Milicic's On the Gram-Schmidt Projection in Normed Spaces. There, Eq. 6 defines the Gram-Schmidt projection of a vector $x\in X$ on the subspace $Y=\text{span}\{y_1\dots y_n\}$ as:
$$ \bar{x} = - \frac{1}{\Gamma(y_1\dots y_n)} \begin{vmatrix} 0 & y_1 & \dots & y_n\\ g(y_1, x) & g(y_1,y_1) & \dots & g(y_1, y_n) \\ \vdots & \vdots & \ddots & \vdots \\ g(y_n, x) & g(y_n,y_1) & \dots & g(y_n, y_n) \end{vmatrix} $$ where $\Gamma(\cdot)$ is the Gram-determinant and $g(\cdot)$ is the semi-inner product. $|\cdot|$ is a determinant sign. Here I do not understand how this calculation can yield a vector. The same notation exists in follow-up papers and as a non-mathematician I am feeling like I am missing some conventions.