Matrix of vectors

Question

For some context, I am learning about the cross product, and the matrix we used as an example was: $ \textbf{V} = \begin{bmatrix} \vec{v_{1}} \\ \vec{v_{2}} \\ \vec{v_{3}} \\ \end{bmatrix} $ which I am told is a 3 x 3 matrix. But since these vectors are in 3 dimensions, wouldn't they also be column vectors? How do we get to a 3 x 3 square from that?

Answer 1

Don't worry, it's just a matter of notation (check at the beginning of your notes/text): if $v^i=(v^1_i,v^2_i,v^3_i)$ (where I write $v_i$ to indicate the $i$-th vector and $v^j$ the component, following you notation -I eliminate the arrow to simplify it-), the matrix $V$ you obtain is \begin{equation} \begin{pmatrix} v^1_1 & v^2_1 & v^3_1 \\ v^1_2 & v^2_2 & v^3_2 \\ v^1_3 & v^2_3 & v^3_3 \\ \end{pmatrix} \end{equation} Otherwise you consider $V^T=[v_1, v_2,v_3]$, and then you list your vectors in column. Hope it help!