I'll say $\vec{a}$ and $\vec{b}$ are collinear if they lies in same line, like (1, 2, 3) and (-1, -2, -3). What should I call two vectors in same direction but with different lengths? like (1, 2, 3) and (2, 4, 6). (Of course they are collinear, but is there any more specific term for them?)
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1$\begingroup$ "codirectional"? $\endgroup$– f5r5e5dJan 16, 2017 at 5:24
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$\begingroup$ "Positive scalar multiple" is the best I can think of. Incidentally, "parallel" usually refers to vectors which are a not-necessarily-positive scalar multiple of each other. One reason that's a more natural notion is that we're often interested in vector spaces whose field of scalars isn't linearly ordered - e.g. vector spaces over $\mathbb{C}$ - so there is no notion of a "positive" scalar. $\endgroup$– Noah SchweberJan 16, 2017 at 17:02
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2 Answers
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You can say (1,2,3) and (2,4,6) are parallel.
As $ \vec{a} = k \vec{b}$
Edit - Difference between parallel and collinear
If two parallel vectors have angle = 0 between them that is both are in same direction but collinear vector lie in same plane they may be parallel or anti parallel that is angle = 180.
answered Jan 16, 2017 at 4:50
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1$\begingroup$ I wonder why this was downvoted. It is an accurate answer, with the possible ambiguity in the case of $(1,2,3)$ and $(-2,-4,-6)$, which are parallel, but which the OP might not consider "in the same direction." $\endgroup$– davidlowryduda ♦Jan 16, 2017 at 4:51
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$\begingroup$ Those who down vote please mention the reason so that I can know my mistake. $\endgroup$ Jan 16, 2017 at 4:55
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3$\begingroup$ I didn't down vote, but I don't see how it answers the question. OP already has a term they can use for vectors $\vec{a}, \vec{b}$ with $\vec{a} = k \vec{b}$ where $k$ may or may not be negative. I just don't see anything in this answer that refers to positive $k$ only (the OP's objective). $\endgroup$– pjs36Jan 16, 2017 at 4:59
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The proper term is that one vector is a positive scalar multiple of the other. Or you could say that they give the same normalised vector (which is the vector obtained by dividing them by their own length). In a more informal discussion you could say the vectors point in the same direction.
answered Jan 16, 2017 at 5:49
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$\begingroup$ I think this answer is a pretty strong indication that there doesn't exist a single term for that relationship between two vectors -- or if there is one, that it's quite localized! $\endgroup$– pjs36Jan 16, 2017 at 6:00