[western50]

1. Are the lines with parametric formulas
Subscript[L, 1][t]={2,3}+ t{-3,5}
and
Subscript[L, 2][t]={2,3}+ t{6,-10}
the same line or different lines? These two formulas are different line because their expressions for parallel vector are not the same, right? I am not sure how I should explain this problem.

[Unknown008]

The simple fact that they don't have the same vector direction says that they are not the same lines. It's pretty similar to the linear equations you learned previously.

If you have two lines with the same gradient, there is a good chance that they are the same line.

So first, you verify the gradient (vector) then you verify the point. If a line has the same gradient (vector) and go through the same points (usually, only one point is required), then the equations represent the same line.

Can you find the vectors of the lines?

[jcaron2]

They're the same line. The parameter t for one line is NOT the same as the t for the other line, but if you include all values for t from -infinity to infinity, they trace out the same path. If you choose any value of t for the second equation L2(t), you can get the same result with the first equation L1(t) by simply using t_1 = -2t_2.

Hopefully that makes sense. I don't think my explanation is a very good one. Let me know if you want clarification.

[jcaron2]

Another way to approach this problem, by the way, is to convert from parametric equations back into good ol' fashioned slope-intercept form if that's what you're more comfortable with.

L1 = {2,3}+ t{-3,5}

This can be rewritten as a system of two equations in x and y:

x = 2 -3t
y = 3 + 5t

Solving the first for t, we get t = -(x-2)/3. Plugging that into the second, we get y = -5/3x + 19/3.

Now you can do the same thing with the second equation for L2, and you'll find you end up with exactly the same slope-intercept-form equation.