What is the Direction of a Zero (Null) Vector?

Question

To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would back it up.

Wikipedia's entries on the zero vector seems to agree with that intuition, but as always, one shouldn't blindly trust Wikipedia: in one place, it is stated that a zero vector "is orthogonal to all other vectors with the same number of components," while in another, it is stated that "two vectors can be considered orthogonal if and only if their dot product is zero, and they have non-null length." Correct me if I am wrong, but these two statements contradict each other.

This question popped up in my head when I heard someone arguing that since a Euclidean vector is defined as a geometric entity that has both a magnitude and direction, and since a zero vector is a vector with length 0, then it is only fitting for a zero vector to have a "direction 0." I am personally inclined to say that a zero vector does not have a particular direction, but as I said, I would like to know if there is a rigorous formulation that would lead to this conclusion.

And to put the question in a less "discussion-inducing" form, is there an agreement on the direction of a Euclidean zero vector?

Answer 1

The zero vector has no particular direction; this is consistent with the fact that it is orthogonal to every other vector. (It doesn't really make sense to say it has "direction 0", since direction is not a magnitude; "direction 0" makes no more sense than "direction 1" or "direction 5.873".)

Alternatively, you could say that it points in every direction, but with zero magnitude, since if you take any vector and multiply it by zero, you get the zero vector. "Every direction" is the same as "no particular direction"; it's just a different way of phrasing things.

This is one flaw with the traditional description of a vector as being a pair consisting of a magnitude and a direction: For the zero vector, the magnitude is zero, but the direction is arbitrary.

Answer 2

Since adding the zero vector to any non-zero vector does not change the direction of the latter, it cannot have a proper direction because of the parallelogram rule :-)

Answer 3

Just a remark: there is some relevant basic algebraic topology here.

As Matt E points out in his answer, the "direction" of a vector $v \in \mathbb{R}^n$ is not a number. What then is it? A natural answer is that a direction is an element of the unit sphere $S^{n-1} = {(x_1,\ldots,x_n) \in \mathbb{R}^n \ | \ x_1^2 + \ldots + x_n^2 = 1 }$.

Then the map, say $D$, which assigns each $v \in \mathbb{R}^n \setminus {0}$ to its direction to its direction $D(v) = \frac{v}{||v||} \in S^{n-1}$ is a deformation retraction. It follows that $\mathbb{R}^n \setminus {0}$ has the same homotopy type as the sphere $S^{n-1}$. In particular, $\mathbb{R}^n \setminus {0}$ does not have the same homotopy as $\mathbb{R}^n$ -- i.e., it is not contractible. This can be taken as a precision of the idea that there is no natural way to extend $D$ so as to be defined at the zero vector. (For that matter, there is evidently no continuous extension of $D$ to $\mathbb{R}^n$, since $D$ is surjective on any deleted neighborhood of $0$.)

Note also that any algebraic geometer will inevitably be reminded of projective space, in which the zero vector must be excluded for similar reasons. In this context, (real) projective $n-1$-space is obtained by failing to distinguish between the directions of $v$ and $-v$. Topologically, this amounts to taking a quotient of $S^{n-1}$ by identifying antipodal points.

Answer 4

I think it is possible for a zero magnitude vector to have direction. Consider a particle moving in a circle at constant speed. Velocity is variable because direction changes. magnitude of acceleration is zero because speed is constant. Acceleration (directed towards center) is variable because direction changes every instant. Clearly acceleration is present with magnitude zero.

Answer 5

First, we need a definition of direction.

How about this one: the direction of a vector $x$ is a vector $u$ of unit length (that is, $|u| = 1$) such that $cx = u$ for some positive real number $c$.

According to the definition, it is clear that if $x = 0$, then there exists no $c$ such that $cx$ is a unit vector.

Also, it is clear that the direction $d(x)$ of a vector $x$ is a function mapping $x$ to a unit vector. For any given $x$, the coefficient $c$ is just $\frac{1}{|x|}$, and so the function is this: $$d(x) = \frac{x}{|x|}$$.

It is clear that the function is not defined at $x = 0$, where it invokes division by zero.

Answer 6

A vector can exist with a zero magnitude and a direction. Express the vector as a scalar times a unit length vector. The unit length vector indicates the direction and the scalar represents the length. Even if the length goes to zero we still have the direction. This becomes obvious if we use polar or spherical coordinates.