So I came across this question: Given vector $\textbf{u} = i+j, \textbf{v} = j+k, \textbf{w} = i+k$. Find the triple scalar product $u(\textbf{v}\times \textbf{w})$.
So I tried to check my notes to see if I can solve it myself. And according to my notes, the triple scalar product of vectors $\textbf{u} = i+j+k, \textbf{v} = i+j+k \text{ and } \textbf{w} = i+j+k$ is the determinant of the $3\times3$ matrix formed by the components of the vector.
If for my case I only have $2$ for each vector, does that mean the matrix is $2\times3$? I am kinda confused.
Those vectors should be interpreted as \begin{align*}u &= 1\cdot\vec{i} + 1\cdot \vec{j}+ 0\cdot\vec{k}\\ v &= 0\cdot\vec{i} + 1\cdot \vec{j} + 1\cdot\vec{k}\\ w &= 1\cdot\vec{i} + 0\cdot \vec{j}+ 1\cdot\vec{k}\end{align*} so that the coefficient matrix we'll be taking the determinant of is $\begin{pmatrix}1&1&0\\0&1&1\\1&0&1\end{pmatrix}$.
The other components are always there, even if their coefficients are zero. These are vectors in $\mathbb{R}^3$, so we use coefficients with respect to the standard basis $\vec{i},\vec{j},\vec{k}$, and that means all three coefficients to express any particular vector.