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Lets say I have a position vector $\vec r$. Is it dimensionless or does it have a dimension of length i.e $[L]$.

Also does the unit vector $\hat r$ have a dimension?

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Vectors do have dimensions. Specifically, the dimension of a vector is (and always must be) the same as the dimension of its components. This also means that al the components of a vector must have the same dimension.

In your example, the position vector $\vec r$ does indeed have units of length.

The vector $\hat r$ is defined as $\hat r = \vec r / |\vec r|$. Since we know that both $\vec r$ and $|\vec r|$ have units of length, we can conclude that $\hat r$ has units $L / L = 1$. In other words, it is dimensionless.

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  • $\begingroup$ So that would mean that the unit vector $\hat r$ too has dimensions of length? $\endgroup$ – Binary Geek Mar 28 '15 at 15:32
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    $\begingroup$ @BinaryGeek see my edit $\endgroup$ – JSQuareD Mar 28 '15 at 15:53
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Yes it does have the dimension $[L]$. You can, however, be more specific and assign a different length dimension to each component. So that the $x$ component has length dimension of $[L_x]$ and analogous for $y$ and $z$. These are called direction dimensions, and can be helpful in dimensional analysis.

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  • $\begingroup$ So that would mean that the unit vector $\hat r$ too has dimensions of length? $\endgroup$ – Binary Geek Mar 28 '15 at 15:32
  • $\begingroup$ No, $\hat r$ has no units, as it is defined as the vector $r$ divided by the magnitude of $r$, which themselves both have the same units and therefore cancel to give a dimensionless unit vector $\endgroup$ – danimal Mar 28 '15 at 15:43
  • $\begingroup$ @Joseph, thanks but I'm not the OP and that's basically what I said :) $\endgroup$ – danimal Mar 28 '15 at 16:03
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I think it should have dimensions. Suppose the question were "what is the unit vector of 10 Newtons force pointing due north?". Then the answer is "1 Newton due north".

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    $\begingroup$ A unit vector is a vector which has a magnitude of exactly 1, the arithmetic unit multiplier. It has no units. $\endgroup$ – Bill N Sep 14 '15 at 19:10
  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. $\endgroup$ – John Rennie Sep 15 '15 at 5:52
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    $\begingroup$ @John I think it is an answer. The question is whether a vector has units, and this answer says it should. It may have quality issues, but it does answer the question. $\endgroup$ – David Z Sep 15 '15 at 6:46

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