Unit Vectors — Question 4

Question 4

The zero vector has no preferred unit direction

Two families of displacement vectors are $\vec A(\epsilon)=\epsilon\hat\imath$ and $\vec B(\epsilon)=\epsilon\hat\jmath$, with $\epsilon>0$ measured in metres. Both approach zero as $\epsilon$ decreases to zero.

1. Normalize each family for every positive $\epsilon$.

2. Can a single unit vector be assigned continuously to the zero vector by this limiting process?

3. A program returns $(1,0)$ whenever a measured vector is exactly zero. What does that output mean, and how should the result be reported?

Question 4 - Solution

The zero vector has no preferred unit direction

(a) Divide before taking the limit.

$$\boxed{\frac{\epsilon\hat i}{|\epsilon\hat i|}=\hat i,\quad\frac{\epsilon\hat j}{|\epsilon\hat j|}=\hat j}\quad(\epsilon>0)$$ $|\vec A|=|\vec B|=\epsilon$, so $\vec A/|\vec A|=\hat\imath$ and $\vec B/|\vec B|=\hat\jmath$ for all $\epsilon>0$. Their magnitudes tend to zero while their unit directions remain different.

(b) Compare approaches to the same endpoint.

$$\lim_{\epsilon\to0^+}\widehat{\vec A(\epsilon)}=\hat i\ne\hat j=\lim_{\epsilon\to0^+}\widehat{\vec B(\epsilon)}$$ No unique continuous assignment is possible: approaching along east gives $\hat\imath$, while approaching along north gives $\hat\jmath$. At exactly zero, the normalization formula is division by zero. The zero vector is well defined, but its direction is not.

(c) Distinguish a software convention from measured direction.

$$|\vec0|=0,\qquad\vec0/0\text{ undefined};\quad(1,0)\text{ is a software convention}$$ The output $(1,0)$ is an arbitrary fallback convention, not a physical eastward direction inferred from zero data. Report zero magnitude and undefined direction, or explicitly mark the returned direction as a placeholder. If measurements are merely small, their directional uncertainty must be assessed separately.