I was reading triple vector product. I saw an expression like this : $${\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{x}&=\mathbf {u} _{y}(\mathbf {v} _{x}\mathbf {w} _{y}-\mathbf {v} _{y}\mathbf {w} _{x})-\mathbf {u} _{z}(\mathbf {v} _{z}\mathbf {w} _{x}-\mathbf {v} _{x}\mathbf {w} _{z}) \\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})+(\mathbf {u} _{x}\mathbf {v} _{x}\mathbf {w} _{x}-\mathbf {u} _{x}\mathbf {v} _{x}\mathbf {w} _{x})\\&=\mathbf {v} _{x}(\mathbf {u} _{x}\mathbf {w} _{x}+\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{x}\mathbf {v} _{x}+\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{x}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{x}\end{aligned}}}$$
In the first term, I was trying to understand first term. From cross product of $\vec v$ and $\vec w$, I get $$\vec v \times \vec w=(v_yw_z-v_zw_y)\hat x+(v_xw_z-v_zw_z)\hat y+(v_xw_y-v_yw_x)\hat z$$
I was thinking determinant of triple vector cross product should look like, I failed to find it in internet. $$\left|\begin{matrix}\hat x & \hat y & \hat z\\\ v_x & v_y & v_z \\\ w_x & w_y & w_z \\\ u_x & u_y & u_z \end{matrix}\right|$$
Is it the correct form? What I think of the expression that is, it's wrong.
In first equation, they wrote only for x coordinate. I can see that they removed $\hat x$, why? For triple vector, they just wrote $u_y$ instead of $\hat z$ and $u_z$ instead of $\hat y$ why?