An object that has direction but no magnitude?

[Mooky]

A scalar has magnitude and no direction; a vector has both direction and magnitude. Is there a name for a physical or mathematical object with direction but no magnitude?

I guess a description of it could be something like

\lim_{\|v\|\to 0} \vec{v}

 

[Nabeshin]

I don't think so. Mostly, when we want to ignore magnitude, we just make the vector a 'unit' vector, i.e. one of norm 1, so it picks up the magnitude of whatever is multiplying it. These vectors are used for indicating direction only.

 

[lbrieda]

As Nabeshin said, this is basically what a unit vector is. You can think of a unit vector as having only a direction since you need to multiply it by the scalar magnitude in order to get some physical meaning.

 

[Simon Bridge]

Second that - however, we call a direction without a magnitude an "angle".
Look at how spherical-polar coords are set up?

It's just not very useful.
The zero-length (r=0) polar vector can technically have any angle, but they all point to the same place: the origin.

 

[Per Oni]

Is there a name for a physical or mathematical object with direction but no magnitude?

I guess a description of it could be something like

\lim_{\|v\|\to 0} \vec{v}

Another discription is: windvane

 

[Mooky]

we call a direction without a magnitude an "angle".

Yes, in the 2D case. But, is there a generalization of the notion of angle to higher dimensions, say $\mathbb{R}^9$?

 

[Zula110100100]

I would think that in R9 it would still be an angle, since in 3d, we still have an angle that spans the three dimensions and can be broken down to 2-2d angles, this could keep happening, so in 4d, you could have an angle that breaks down to 3 2d angles? Thats my guess anyway

 

[Simon Bridge]

+1 to that: of course it is generalizable to more than 2D.

The angle between two nD vectors would be the inverse-cosine of their scalar product divided by the product of their magnitudes. The nD equivalent of saying "over thataway" and pointing.

It is often useful to represent the angular position in nD as a set of n-1 orthogonal angles, as in 3D spherical-polar coordinates. In the same way you can define n-1 unit angles on an n-ball.

The nD equivalent of saying "look to the left of that and a bit down from the other thing".

 

[Redbelly98]

I don't think so. Mostly, when we want to ignore magnitude, we just make the vector a 'unit' vector, i.e. one of norm 1, so it picks up the magnitude of whatever is multiplying it. These vectors are used for indicating direction only.

As Nabeshin said, this is basically what a unit vector is. You can think of a unit vector as having only a direction since you need to multiply it by the scalar magnitude in order to get some physical meaning.

But a unit vector does have a magnitude: it's 1.

 

[Simon Bridge]

Which is why we can use angles to indicate relative direction.