Q.Find the cartesian equation of the line which passes through (1, 2, 3) and parallel to the line (x+3)/3 = (y-4)/5 = (z+8)/6.

Parallel lines share the same direction ratios — read those off the given line and write the symmetric form through the new point.

Given line: $\dfrac{x+3}{3} = \dfrac{y-4}{5} = \dfrac{z+8}{6}$, with direction ratios $(3,5,6)$.

Key fact: two lines are parallel iff they have proportional direction ratios. So the required line, being parallel to the given one, also has direction ratios $(3,5,6)$.

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