Are two vectors $\mathbf x, \mathbf y$ considered parallel if $\mathbf x=- \mathbf y$? In my linear algebra class we discussed parallel vectors. However, I've come upon a homework question that has really confused me. I tried asking the professor, but he didn't understand what I was trying to say. Here is the question:
Determine whether the vectors emanating from the origin and terminating at the following points are parallel. Points: $(5, -6, 7)$ and $(-5, 6, -7)$.
Clearly, $(-5, 6, -7)$ is a multiple of $(5, -6, 7)$. Observe that $(-1)(-5, 6, -7)=(5, -6, 7)$. Also, even though we haven't discussed cross products I know the concept from calculus which says that when $\mathbf u \times \mathbf v= \mathbf 0$ the two vectors are parallel, so
$$\begin{vmatrix} \vec{i}&\vec{j}&\vec{k}\\ 5&-6&7\\-5&6&-7\end{vmatrix}=\vec{i}(42-42)-\vec{j}(-35+35)+\vec{k}(30-30)=\mathbf0$$
So my logic tells me that the two vectors are parallel, but I'm sure there must be a flaw with my understanding because the answer given says that these two vectors are not parallel. Can somebody please clarify or direct me to a similar post that can clear up my confusion? 
Thanks a lot in advance!
(By the way the homework has already been submitted and returned to me, but this question was not one of the questions selected for grading)
 A: I would say, maybe you can write the as the term "anti-parallel".
Although even that, would just be a part of "parallel", more like a special case of it where the direction of the two vectors is in opposite direction.
I would go with the fact that the solution might be incorrect, unless the author really wants you to state "anti-parallel" instead of just "parallel".
http://en.wikipedia.org/wiki/Antiparallel_%28mathematics%29
A: Most writers define two vectors $\mathbf{x}$ and $\mathbf{y}$ to be parallel if there exists $k\neq0$ such that $\mathbf{x}=k\mathbf{y}$. By this definition then, the answer to your title question is "yes": Two vectors $\mathbf{x}$ and $\mathbf{y}$ are parallel if $\mathbf{x}=-\mathbf{y}$.

(Note though that a minority of writers define two vectors $\mathbf{x}$ and $\mathbf{y}$ to be parallel if there exists $k>0$ such that $\mathbf{x}=k\mathbf{y}$ and anti-parallel if there exists $k<0$ such that $\mathbf{x}=k\mathbf{y}$. By these definitions then, the answer to your title question is "no": Two vectors $\mathbf{x}$ and $\mathbf{y}$ are not parallel if $\mathbf{x}=-\mathbf{y}$; instead, they are anti-parallel.)
