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Parametric equations for the line through {1,2,3} parallel to the vector {5,6,-4} are
x=1+ 5t,
y=2+ 6t, and
z=3- 4t.
A friend taking the old-fashioned course tells you that the xyz-equations for this line are
(x - 1)/5=(y - 2)/6=(z - 3)/-4.
Your friend is right.
Why?

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