= Solution
Decompose $\mathbf p$ into its <orthogonal projections> parallel and perpendicular to $\hat{\mathbf a}$. Since $\mathbf p\times\hat{\mathbf a}$ removes the parallel component,
$$
|\mathbf c|=|\mathbf p\times\hat{\mathbf a}|
=|\mathbf p-(\mathbf p\cdot\hat{\mathbf a})\hat{\mathbf a}|.
$$
The last <vector> is the displacement to the point of the <straight line> nearest the origin. Hence
$$
\boxed{c\text{ is the perpendicular distance from the origin to the line}}.
$$
The original PDF has $c=|\mathbf c|$, not the squared magnitude inserted by the OCR.
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