Initial point on the origin
v1 = (−1, 2, 3), v2 = (2,−4,−6) and v3 = (−3, 6, 0)
How do we do this?
I know they are linearly dependent as 2v1 + v2 + 0v3 = 0
No, saying that three vectors are linearly dependent means they lie in the same plane so it is still possible that they lie on the same line.
If they were independent, then they could not be on the same line but knowing that they are dependent doesn't tell you whether they lie on the same line or not. Do as Plato suggested- are they all multiples of one another?
I hate to add to a "once and for all" but the initial post said that these were vectors with "Initial point on the origin". So it is sufficient to determine whether, say, (2, -4, -6) and (-3, 6, 0) are multiples of (-1, 2, 3).
Of course (2, -4, -6)= -2(-1, 2, 3) so that is a multiple. But (-3, 6, 0) is NOT a multiple of (-1, 2, 3). -3= 3(-1) and 6= 3(2) but 0 is NOT 3 (3). These vectors are do NOT lie on a single line.
One more time, adam leeds, showing that three vectors are "dependent", that is showing that 2v1 + v2 + 0v3 = 0, only shows they lie in the same plane, not that they along the same line.
It is also, of course, true, as Plato says, that $\displaystyle v_1- v_2= (-3, 6, 9)$ and [tex]v_3- v_2= (-5, 10, 6) are not multiples of one another. -5= (5/3)(-3) and 10= (5/3)(6) but 6is NOT equal to (5/3)(9)