# Hi i know this is a really really simple question but it has me confus

Hi i know this is a really really simple question but it has me confused.
I want to calculate the cross product of two vectors
$$\displaystyle\vec{{{a}}}\times\vec{{{r}}}$$
The vectors are given by
$$\displaystyle\vec{{{a}}}={a}\vec{{{z}}},\ \vec{{{r}}}={x}\vec{{{x}}}+{y}\vec{{{y}}}+{z}\vec{{{z}}}$$
The vector $$\displaystyle\vec{{{r}}}$$ is the radius vector in cartesian coordinares.
I want to calculate the cross product in cylindrical coordinates, so I need to write $$\displaystyle\vec{{{r}}}$$ in this coordinate system.
The cross product in cartesian coordinates is
$$\displaystyle\vec{{{a}}}\times\vec{{{r}}}=-{a}{y}\vec{{{x}}}+{a}{x}\vec{{{y}}}$$,
however how can we do this in cylindrical coordinates?

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Donald Proulx
The radius vector $$\displaystyle\vec{{{r}}}$$ in cylindrical coordinates is $$\displaystyle\vec{{{r}}}={p}\vec{{{p}}}+{z}\vec{{{z}}}$$. Calculating the cross-product is then just a matter of vector algebra:
$$\displaystyle\vec{{{a}}}\times\vec{{{r}}}={a}\vec{{{z}}}\times{\left({p}\vec{{{p}}}+{z}\vec{{{z}}}\right)}$$
$$\displaystyle={a}{\left({p}{\left(\vec{{{z}}}\times\vec{{{p}}}\right)}+{z}{\left(\vec{{{z}}}\times\vec{{{z}}}\right)}\right)}$$
$$\displaystyle={a}{p}{\left(\vec{{{z}}}\times\vec{{{p}}}\right)}$$
$$\displaystyle={a}{p}\vec{{\phi}}$$,
where in the last line we've used the orthonormality of the triad $$\displaystyle{\left\lbrace\vec{{{p}}},\vec{{\phi}},\vec{{{z}}}\right\rbrace}$$