Yes, there is a version of the equation that doesn't require vectors. One way to write the equation of the line of intersection is to use the point-slope form, which is given by:

L = P + m(V),

where P is a point on the line, V is a second point on the line, and m is the slope of the line, which can be calculated as:

m = (V - P)/ ||V - P||

Using this form, we can substitute the values of P and V that we found earlier:

L = (1, -1, 1/3) + m((14, -4, -18) - (1, -1, 1/3))

This equation gives the equation of the line of intersection without using vectors.
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