[Math] About Null vector orthogonal or Parallel to any other vector

linear algebravectors

I have one basic doubt regarding Null vector.

By definition of Dot product if $\vec{a}$ is any vector and $\vec{b}$ is Null vector then its obvious that $$\vec{a}\cdot\vec{b}=0 \tag{1}$$ that is a Null vector is Orthogonal to any vector.

Similarly By definition of cross product if $\vec{a}$ is any vector and $\vec{b}$ is Null vector then its obvious that $$\vec{a} \times\vec{b}=\vec0 \tag{2}$$ that is a Null vector is parallel to any vector.

But by definition of null vector, Null vector is a vector with zero magnitude and no Specific direction (Points in different directions viewing it as circle with zero radius). So practically how can $(1)$ and $(2)$ be true?

Best Answer

That's a "problem" that arises often if some Null (may it be a number, a vector, the empty set or whatever) is involved. Many properties hold at once for the Null (so your statements (1) and (2)) or are obviously not satisfied (like $0\in\mathbb R$ can't be inverted).

So, when you are going to define some mathematical terms (e.g. "orthogonal" or "parallel"), you have to decide:

  • Define it by an easy to check property (like $a$ is orthogonal to $b$ if and only if $a\cdot b = 0$)

or

  • Define it with respect to some vivid picture that somewhat generalises the everyday experience (like $a$ is parallel to $b$ if and only if they point in the same direction).

As mathematics deals with rather general terms, in general the first option is preferred (unless you work onlay with simple structures, where the second option is easier to grasp).

This approach works very well, but with the consequence that in simple cases the statements sound a bit odd if compared to the everyday experience. (As the Null vector is orthogonal to itself, or - regarding group theory - as the trivial permutation is a symmetry.)

In your case you have to consider the given definition of "orthogonal" and "parallel".

  • If vectors $a$ and $b$ are orthogonal to eachother if and only if $a\cdot b = 0$ per definition, then the Null vector is orthogonal to every other vector and the case is closed.
  • If vectors $a$ and $b$ are orthogonal to eachother if and only if they define a specific direction and these directions interact in a special sense, then you have to alter your used theorem to "If $a\cdot b = 0$ then either $a$ is orthogonal to $b$ or $a$ or $b$ is the Null vector."

(As you see, the second more vivid definition causes theorems where special cases are excluded, which is very annoying when working with a bunch of such theorems. You have always to check if there is maybe some Null.)

In short: It all depends on the used definition of some mathematical term.

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