# Zero vector perpendicular

1. Oct 26, 2006

### jbusc

This is an interesting question that my friend had...

Is the zero vector perpendicular to all other vectors?

We know that the zero vector is orthogonal to all other vectors; I verified this in both "Linear Algebra and it's Applications" by Strang and "Calculus" by Stewart; both are exceedingly clear about this.

Now while this may be pedantic, does this translate into "perpendicularity"?

If perpendicularity means the same as orthogonality (vanishing of the inner or dot product) then the argument is over, and the zero vector is perpendicular to all vectors. Mathworld (http://mathworld.wolfram.com/Perpendicular.html) agrees with this sentiment.

While we typically think of perpendicularity as equivalent to orthogonality, however, there's the argument that orthogonality is some kind of generalization of perpendicularity. In this argument perpendicularity can be defined as vectors which form 90 degree angles (and thus only makes sense in Euclidean n-space R^n), whereas orthogonality is the vanishing of the inner product and so is valid in any inner product space. Since we cannot claim that the zero vector forms any angle at all, it cannot form a 90 degree angle, and then could not be perpendicular to anything.

However, even if the case that "perpendicularity" is defined as forming 90 degree angles, if it is to be a generalization of orthogonality, then in regular R^n space, vectors should be perpendicular iff they are orthogonal as well (otherwise, it is not much of a generalization if the generalized version does not degenerate fully into the less general version in the basic case). Therefore, we again have the zero vector perpendicular to all vectors.

It seems like part of the problem is lack of a rigorous and standardized definition of perpendicular. Since perpendicularity is not a terribly important concept compared to orthogonality, Strang for example barely mentions perpendicularity at all.

2. Oct 27, 2006

### tandoorichicken

http://garnet.acns.fsu.edu/~jflake/math/GeomSp/GSPerp.html

According to this definition of perpendicularity, A line intersecting the zero vector reflected about the zero vectore results in the same line. A line reflected but not intersecting the zero vector results in a parallel line. Therefore only vectors that intersect with a given zero vector are perpendicular to it.

Last edited by a moderator: Apr 22, 2017
3. Oct 28, 2006

### marcmtlca

I agree with mathworld.

4. Oct 28, 2006

### JasonRox

Same here.

I think this is over analyzing it.

The zero vector is orthogonal to every vector. Case closed.

5. Oct 29, 2006

### jbusc

I agree it's overanalyzing it as well. But my friend was marked off points for it on an exam, and the TA insisted that the zero vector did not count as perpendicular. The ensuing discussion (and disagreement) degenerated into this. :)

Personally, I thought the issue was settled by mathworld once and for all also.

6. Oct 29, 2006

### Office_Shredder

Staff Emeritus
If you take the cross product of parallel vectors, you get the zero vector, which means the zero vector is supposed to perpendicular to those.

7. Oct 29, 2006

### matt grime

Mathworld cannot settle the issue as far as the convention of your course is concerned.

8. Oct 29, 2006

### mathwonk

apparently your TA and prof have given a special definition of perpendicular, different from the rest of the world, and you need to be aware of that when taking their tests.

they are wrong however, if you want to go with what the entire remainder of humanity means by perpendicular. [i have asked everyone in the world, and so i know.]

9. Oct 31, 2006

### jbusc

Well it's not my course...it's not even my university. It's primarily a matter of the fact that neither the professor nor the TA provided any kind of indication that it didn't; it's just that they didn't anticipate that answer on the exam and the TA claiming that it wasn't that case, and my poor friend (too smart for his own good, knowing the mathematical definition of orthogonal and such) getting shafted.

It's not a particularly small issue either; it's 10% of an exam which is a third of the course grade, roughly 3.3% of the final grade affected by this one problem.