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When I draw two collinear vectors, say $a(2,4,6)$ and $b(4,8,12)$, using a simulator, they end up being the same line and starting from the origin. They are not separated in space. The usual images of collinear vectors in textbooks show them as being separated in space.

Link to the simulator.

How do I visualise or simulate two vectors which are parallel and separated in the 3-dimensional / 2-dimensional space and not starting from the origin always?

An equivalent statement in the co-ordinate system would be, say, $y=2x$ and $y=2x + 2$ which are parallel lines. They don’t have to pass through the origin and are separated in the plane.

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I perceive a confusion between Cartesian notation of a point in space(3D) & Cartesian notation of a vector. In simulator you are using Cartesian notation of a vector that that represents a line joining origin and given point. That’s how we should look at vectors specified in Cartesian coordinates. See: http://www.maths.usyd.edu.au/u/MOW/vectors/vectors-7/v-7-2.html

If vector V is denoted by $(x,y,z)$ then $V=x \bar(i) + y \bar(j) + z \bar(k)$ Where $\bar(i), \bar(j), \bar(k)$ are unit vectors along Cartesian axes $X,Y,Z$ Also. $V=(x,y,z)$ is actually a vector of magnitude $|V|=\sqrt{x^2+y^2+z^2} $and along the direction of a unit vector $\bar(v)$ whose Cartesian representation is $\bar(v) = ( x/|V|,y/|V|,z/|V|)$ Let’s look at your two vectors $a=(2,4,6), b=(4,8,12)$ They differ only in magnitude $2|a|=|b| $but have identical unit vector, $\bar(a) , \bar(b) $and therefore they are collinear. To plot vector a, you draw a line from origin along the direction of unit vector $\bar(a)$ of length $|a| $& similarly to plot vector $b$ you draw a line from origin along the direction of unit vector $\bar(a)==\bar(b)$ whose length is $|b| = 2 |a|$ they differ only in magnitude, their end tips are different,their unit vector is same. For this reason they appear along same line on simulator plot.

Two vectors are said to be parallel if they have identical direction that is identical unit vector. See: https://www.onlinemathlearning.com/parallel-vectors.html The point of application of a vector is irrelevant. The question of parallel vectors that you are kin to represent are vector having same direction but different point of application. If you specify point of application then only we can represent two parallel vectors. Hope it clarifies.

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