Can I use algebraic equality operation for parametric equations of lines? I am working on below question from a linear algebra book that I am studying,
Suppose $X = X_0 + tV$ and $Y = Y_0 + sW$ are two parametric representations of the same line $l$ in $R^n$.
a. Show that there is a scalar $t_0$ so that $Y_0 = X_0 + t_0V$.
b. Show that $V$ and $W$ are parallel.
Because both equations are representation of the same line, can I claim that $X=Y$ or $X_0 + tV = Y_0 + sW$ in algebraic sense? Assuming it is possible I can prove (a) like this:
$$X_0 + tV = Y_0 + sW$$
$$Y_0 = X_0 + tV - sW$$
Since $W$ and $V$ are direction vectors of the same line, they are parallel, and I can say $W=cV$,
$$Y_0 = X_0 + tV - scV$$
$$Y_0 = X_0 + V(t - sc)$$
$t$, $s$, and $c$ are all scalars, so $t_0=t-sc$,
$$Y_0 = X_0 + t_0V$$
Here is my second question in regards to these proofs:
Using the same assumption that I can say $X=Y$ and the (a), I can prove (b) like this:
$$X_0 + tV = Y_0 + sW$$
Using (a) ($Y_0 = X_0 + t_0V$):
$$X_0 + tV = X_0 + t_0V + sW$$
$$W = \frac{(tV - t_0V)}{s}$$
$$W = V\frac{t-t_0}{s}$$
Which means $W$ is a scalar multiple of $V$ and are parallel.
When I am trying to prove (b) I am using my result from (a) [ $Y_0=X_0+t_0V$ ] which is concluded by the assumption that both vector directions $W$ and $V$ are parallel and infact in (b) I am trying to prove the same assumption that I made in (a) (that they are parallel). I am very new to this stuff and don't know what technical term is used to describe this situation, but it kind of looks like chicken and eggs problem to me. Is there anything wrong with the solutions that I provided?
 A: *

*Can we claim that $X=Y$? Not in general.

$X=X_0+tV$ and $Y=Y_0+sW$ are arbitrary points on the same line determined by the values of $t,s$. If you take any point with position vector $A$ on the line, then $\exists s_1,t_1\in\Bbb R$ such that$$A=X_0+t_1V=Y_0+s_1W$$i.e. $X$ and $Y$ are equal for some pairs $s,t$ that describe the same point on the line and unequal for other pairs that describe different points.


*In proving $(a)$, you have used the fact that $V,W$ are parallel because they are the direction vectors of the same line. This is not a rigorous proof, and you were supposed to prove it in $(b)$ so the question probably expects you to prove $(a)$ without using the fact that $V,W$ are parallel.


For proving $(a)$ without this assumption, note that for $s=0$, we get $Y=Y_0$ is a point on the line. Thus $\exists t_0\in\Bbb R$ such that$$Y_0=X_0+t_0V$$since $X_0+tV$ also describes points on the same line.

For $(b)$, slightly alter your proof like this: Take $s_1\in\Bbb R-\{0\}$. Then $\exists t_1\in\Bbb R$ such that$$X_0+t_1V=Y_0+s_1W$$Now substitute for $Y_0$ from $(a)$ to get $W=\left(\frac{t_1-t_0}{s_1}\right)V$. Note that $t_1\ne t_0$ since $W$ is not zero vector. Also note why taking $s_1\ne0$ was necessary. Thus $W$ is parallel to $V$.
