furajat4h

2022-09-22

For some context, I am learning about the cross product, and the matrix we used as an example was: $\mathbf{\text{V}}=\left[\begin{array}{c}\stackrel{\to }{{v}_{1}}\\ \stackrel{\to }{{v}_{2}}\\ \stackrel{\to }{{v}_{3}}\end{array}\right]$ which I am told is a $3×3$ matrix. But since these vectors are in 3 dimensions, wouldn't they also be column vectors? How do we get to a $3×3$ square from that?

efterynzl

Don't worry, it's just a matter of notation : if ${v}^{i}=\left({v}_{i}^{1},{v}_{i}^{2},{v}_{i}^{3}\right)$ (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
$\left(\begin{array}{ccc}{v}_{1}^{1}& {v}_{1}^{2}& {v}_{1}^{3}\\ {v}_{2}^{1}& {v}_{2}^{2}& {v}_{2}^{3}\\ {v}_{3}^{1}& {v}_{3}^{2}& {v}_{3}^{3}\end{array}\right)$
Otherwise you consider ${V}^{T}=\left[{v}_{1},{v}_{2},{v}_{3}\right]$, and then you list your vectors in column. Hope it help!

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