![]() Well, perpendicular to both means that it must have the formĮxpression you've written down is exactly the formula for that $z$ component. Then $W = U \times V$ is a vector in 3-space perpendicular to both (but possibly zero!). If you have a pair of vectors $u$ adn $v$ in the plane, you can append a $0$ to each one, so if $u = (a, b)$, you get $U = (a, b, 0)$, and similarly for $v$. ![]() The ordinary cross product takes a pair of nonzero vectors in 3-space, $u$ and $v$, and produces a new vector $w = u \times v$ whose length is the area of the parallelogram spanned by $u$ and $v$ (hence the length may be zero), and whose direction (if $w$ is nonzero) is perpendicular to $u$ and $v$, and has the property that $u, v, w$ (in that order) form a right-hand-oriented basis of 3-space. The "vector cross product in 2D" is neither a product (in the strict sense), nor a "vector", nor is it really in 2D.
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