Intuitively, what makes two vector parallel

Question

I have heard explanations such as it is when the cross product equals zero or that it is when one is a scalars multiple of the other but I have not seen an intuitive explanation. Is it when two vectors have the same magnitude and direction?

Answer 1

Two vectors do not need to have the same magnitude to be parallel.

Intuitively, two vectors are parallel if, when you place them on top of eachother, they form one single line. Meaning, they can have the same direction or opposite direction. This also means that if they are not on top of eachother, they will never intersect.

Be aware here, that vectors are not defined by their starting point in any way. A vector with some direction and magnitude is the same vector even if you move its starting point from say $(0,0)$ to $(5, 2)$.

What happens with a vector when you multiply it by a scalar, is that it changes magnitude. It preserves its direction, but simply changes size.

Now, if you multiply it by a negative number, it will have turned around, and is now facing the other way. This should be intuitive because for instance movement, is a vector. Moving $-5m$ in one direction, means moving $5m$ in the opposite direction.

Answer 2

"Magnitude", "direction" make sense only if your vector space has an inner product. Cross product only makes sense, if your vector space is 3-dimensional and has an inner product.

Geometrically speaking, two vectors are parallel, if they have the same "direction" (or opposite direction), regardless of "magnitudes".

This construction makes sense mathematically even if your vector space has no inner product, because if $x$ and $y$ are elements of a vector space $V$ over a field $\mathbb{F}$, then they are parallel if and only if there exists an $\alpha\in\mathbb{F}$ number so that $$ y=\alpha x, $$ and as you can see, no inner product was needed.

Of course, if your space is 3-dimensional, real and has an inner product, then one way to check if two vectors, $x$ and $y$ are parallel is that the cross product $x\times y$ vanishes, because if $x$ and $y$ are parallel, then $y=\alpha x$ and then $$ x\times y=x\times(\alpha x)=\alpha (x\times x)=0, $$ since the cross product is an alternating bilinear map, it is always zero, when you plug in the same vector twice into it.

Answer 3

Parallel vectors have a constant minimum distance between them in Euclidean geometry, a single minimum distance between them in skewed orientation of Hyperbolic geometry, a single maximum distance between them in intersecting Elliptic geometry.