Mark44
Mentor
It is formatted very professionally, but the overall structure should be improved. I won't be able to look at this very closely for a few hours, but here are some initial thoughts.

1. This is an if and only if proof, so it should have two parts.
a. u, v, and w are coplanar ==> u, v, and w are linearly dependent.
b. u, v, and w are linearly dependent ==> u, v, and w are coplanar

2. In each of the two parts above, your first line should be the hypothesis (the part you are assuming is true - the part before ==>). Using logic and reasoning, you should get to the last line, which should be your conclusion (the part after ==>).

3. In 1a above, you need to examine two cases of coplanar vectors: one or more of the vectors are collinear; none of the vectors are collinear. You sort of did this, but there are some holes in your logic.

4. In your argument to show linear dependence you have
u = sv + tw
Then you say "Slightly rearranging equation (1), ..." followed by two more lines.

Instead of saying what you're going to do (since it should be obvious), just do it.
u = sv + tw
<==> u - sv - tw = 0
These two equations are obviously equivalent since all I did was subtract (sv + tw) from both sides. Don't belabor baby steps. Your instructor will get it.

5. The last equation above shows that u, v, and w are linearly dependent, by definition. Any time the equation au + bv + cw = 0 has a nontrivial solution, the vectors are linearly dependent. In this case, a = 1, b = -s, and c = -t. By nontrivial, I mean any set of constants that are not all zero.

6. You have three sections that start "Proof:". There should be only two of these - one for the statement in 1a and one for the statement in 1b.
At the very top you should state the thing you are going to prove.
Vectors u, v, and w are coplanar if and only if they are linearly dependent.
Under that you should have the "only if" part:
If u, v, and w are linearly dependent, then they are coplanar.
Proof: ....

Then the "if" part:
If u, v, and w are coplanar, they are linearly dependent.
Proof: ...

This is the usual ordering of these two parts, but as long as you prove both, the ordering doesn't matter.

ok, thanks...basically, my proof is correct, there are just a few formatting errors right..I will fix these things right away...but apart from that, there aren't any solution errors on my part..correct? Again, thank you..

Mark44
Mentor
It's more than just formatting errors - there are some problems with the structure, as noted above.

thanks!