Yes, there is a version of the equation of the line of intersection that doesn't require vectors. This version of the equation is called the parametric form of the line. In the parametric form, the equation of the line can be written as:

x = x1 + t(x2 - x1)
y = y1 + t(y2 - y1)
z = z1 + t(z2 - z1)

where (x1, y1, z1) and (x2, y2, z2) are two points on the line, and t is a real number.

To find two points on the line of intersection, we can use the two points that we found in the previous solution, one point from each plane. For example, we could use the point (1, -1, 1/3) and the point (3, 0, 1), which lies on both plane 1 and plane 2.

We can then plug these two points into the parametric form of the line to get the equation of the line of intersection:

x = 1 + t(3 - 1) = 1 + 2t
y = -1 + t(0 - (-1)) = -1 + t
z = 1/3 + t(1 - 1/3) = 1/3 + 2/3t

This is the equation of the line of intersection in parametric form, without the use of vectors.
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