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I can't understand the difference between the two.

The definitions are as written in textbook:

Parallel vectors are vectors which have same or parallel support. They can have equal or unequal magnitudes and their directions may be same or opposite.

Two vectors are collinear if they have the same direction or are parallel or anti-parallel. They can be expressed in the form a= k b where a and b are vectors and ' k ' is a scalar quantity.

devang singhi
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    What is your definition for parallel vectors ? – Balloon Dec 24 '15 at 13:29
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    They both are the same thing!! – Jasser Dec 24 '15 at 13:36
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    Unless otherwise stated, a vector is considered free, that is not localized, so @Jasser is right. – Tony Piccolo Dec 24 '15 at 17:11
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    @Tony: In linear algebra (where vectors are based at zero) I agree with you, but in (e.g.) geometry, calculus, and mechanics, one usually (always?) wants to keep track of a vector's basepoint, as well. In my experience, tacit convention depends very much on context. Not saying my interpretation is correct and yours isn't, only that we need the OP's definition of "collinear" to be certain. – Andrew D. Hwang Dec 26 '15 at 14:46
  • @AndrewD.Hwang: I agree but think OP has not in mind a precise definition of collinear, otherwise would have given in the post. – Tony Piccolo Dec 26 '15 at 15:20
  • Thank you for the edit. Now read 2.2 on p. 33 of this source searching for free vectors after linking. – Tony Piccolo Dec 26 '15 at 15:57
  • What does "support" mean here? As far as I know it's not a standard term in pure linear algebra, and the definitions I looked up online don't seem to apply. – epimorphic Dec 26 '15 at 16:25
  • @devang: Thank you for adding explanations regarding "parallel" and "collinear". Unfortunately, it still seems there's not enough information in the question to give a definitive answer. Could you please add to your question 1. Your definition of a vector (particularly, do you distinguish "the same displacement located at two different points"?), and 2. Why you have reason to think "parallel" and "collinear" are, or are not, the same concept? Is it possible that "collinear" vectors necessarily (by definition) have the same location and "parallel" vectors need not have the same location? – Andrew D. Hwang Dec 27 '15 at 15:47

1 Answers1

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$\newcommand{\Reals}{\mathbf{R}}$In some settings, a vector in $\Reals^{n}$ comprises both a "tail" or "location" $p$ in $\Reals^{n}$, and a "displacement" $v$ in $\Reals^{n}$. The ordered pair $(p, v)$ is usually depicted as an arrow from $p$ to $p + v$.

If this is the setting of your question, the vectors $(p_{1}, v_{1})$ and $(p_{2}, v_{2})$ are:

  • Parallel if $v_{1}$ and $v_{2}$ are proportional, i.e., if there exist scalars $k_{1}$ and $k_{2}$, not both zero, such that $k_{1} v_{1} + k_{2} v_{2} = \mathbf{0}$.

  • Collinear if they are parallel and in addition each displacement is proportional to the displacement $p_{2} - p_{1}$ between the vectors' locations, i.e., the arrows representing the two vector lie on a line in $\Reals^{n}$.

In the diagram, all the vectors are (mutually) parallel, but not all are collinear. The blue vectors, for example, are mutually collinear, all lying along the dashed line.

Parallel versus collinear vectors

Andrew D. Hwang
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  • Your definition seems incorrect according to Definition 2 of this article. – tryingtobeastoic Mar 14 '21 at 03:50
  • Vectors don't have such property as starting point, do they? They are all assumed to start from origin point in the direction of x, y and z, so that the $\vec{V}$ is collinear to the $\vec{C}$ if there a point n exists where $\vec{V} = n * \vec{C}$ – IC_ May 08 '21 at 16:31
  • As noted in the comments below the question, in linear algebra, vectors are based at the origin. In other contexts, vectors can be based at different points. The question was probably closed because no definition of parallel was ever provided. This answer was based on an assumption about intent, but specifies how the answer would change based on the question's intent. The OP's use of the term "parallel support" did (and does) suggest to me vectors where the location of the tail matters. – Andrew D. Hwang May 08 '21 at 17:48