Determine if u and v are parallel

Question

Determine if u and v are parallel

$u = <3, -6, 3>$, $v = <-1, 2, -1>$

So i seen a formula that says there is a$ "k"$ such that it if one is a multiple of the other. Is there a number$ k$ such that $ u = kv$?

So with my guess i assumed to do this

$3 = -1k$, $-6 = 2k$, $3 = -1k$

so $k = -3, -3, -3$

I checked the final answer and $u = -3v$

Can someone explain this formula to me. Is each point suppose to be the same in order for it to be parallel. Like if $k = -3, -3, 2$ it wouldn't be parallel because there is a $-3$, $-3$ and $2$, and $2$ is different from $-3$?

Answer 1

To make it easier to demonstrate, let's consider 2 dimensional vectors instead, and let's visualize them in 2d space:

As you can see, if we multiply each of the components of $v$ by $k = -2$, we get the corresponding components of $u$. In fact, a vector is parallel to $u$ if and only if it has this property.

Equivalently, the vectors that are parallel to $u$ are precisely those vectors along the green line, that is, whose coordinates satisfy the equation $2y = x$. And solutions $(x, y)$ to this equation have the property that if you multiply $x$ and $y$ by the same constant $k$, you get another solution.

That, in a nutshell, is what's going on with your formula.

Answer 2

I would think that the vectors are parallel unless they are collinear (i.e. For example, they have the same initial point). By changing the multiple of one of the coordinates, you are changing the slope in one of the dimensions, and the vectors are no longer parallel.