Term for: “equal and opposite” and “sum to zero” (physics)

Question

I'm looking for a word or term that encompasses both of the following conditions:

• equal and opposite
  • AND
• sum to zero

I am looking specifically at a term for a pairing relationship in the context of physics.

Here are some examples:

NUMBER

• ( +5 ) <> ( -5 )
  • equal magnitude
  • opposite sign
  • ( +5 ) ADD ( -5 ) sums to zero

VECTOR / FORCE

• ( magnitude 5 angle 0 degrees ) <> ( magnitude 5 angle 180 degrees )
  • equal magnitude
  • opposite direction
  • ( magnitude 5 angle 0 degrees ) ADD ( magnitude 5 angle 180 degrees ) sums to zero

ELECTRIC CHARGE

• ( 2+ ) <> ( 2- )
  • equal magnitude
  • opposite charge sign
  • ( 2+ ) ADD ( 2- ) sums to zero

Thus far, I've found the terms:

equilibrant : a force capable of balancing another force and producing equilibrium

~ but this is for forces only

anti-parallel : In a Euclidean space, two directed line segments, often called vectors in applied mathematics, are antiparallel, if they are supported by parallel lines and have opposite directions

~ but this is for vectors only and does not strictly mean equal

---

What may help to convey exactly what I am looking for is to consider the pairing relationship from the "opposite end" - let me express my examples again from this point of view:

NUMBER

• we start with number ( 0 )
  • we split ( 0 ) into
  • ( +5 ) <> ( -5 )

VECTOR / FORCE

• we start with vector/force ( magnitude 0 )
  • we split ( magnitude 0 ) into
  • ( magnitude 5 angle 0 degrees ) <> ( magnitude 5 angle 180 degrees )

ELECTRIC CHARGE

• we start with charge ( 0 )
  • we split charge ( 0 ) into
  • ( 2+ ) <> ( 2- )

Answer 1

Additive inverse.

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number),[1] sign change, and negation.[2] For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself.

Vectors -

All the following examples are in fact abelian groups:

Complex numbers: −(a + bi)  =  (−a) + (−b)i. On the complex plane, this operation rotates a complex number 180 degrees around the origin (see the image above).

Addition of real- and complex-valued functions: here, the additive inverse > of a function f is the function −f defined by (−f )(x) = − f (x) , for all x, such that f + (−f ) = o , the zero function ( o(x) = 0 for all x ).

More generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the identity element of this group):

Sequences, matrices and nets are also special kinds of functions.

In a vector space the additive inverse −v is often called the opposite vector of v; it has the same magnitude as the original and opposite direction. Additive inversion corresponds to scalar multiplication by −1. For Euclidean space, it is point reflection in the origin. Vectors in exactly opposite directions (multiplied to negative numbers) are sometimes referred to as antiparallel.

vector space-valued functions (not necessarily linear),

In modular arithmetic, the modular additive inverse of x is also defined: it is the number a such that a + x ≡ 0 (mod n). This additive inverse always exists. For example, the inverse of 3 modulo 11 is 8 because it is the solution to 3 + x ≡ 0 (mod 11).

From Wikipedia - Additive inverse

Answer 2

How about just saying "balanced"? From Oxford Dictionaries:

A condition in which different elements are equal or in the correct proportions.

If you'd like something that conveys the idea that they are equivalent more directly, I'd suggest symmetric:

Made up of exactly similar parts facing each other or around an axis; showing symmetry.

Or in this case, you might even want to say antisymmetric:

Unaltered in magnitude but changed in sign by exchange of two variables or by a particular symmetry operation.

Answer 3

This is the best I can think of: mutually counterbalancing

mutual: directed by each toward the other

counterbalance: to have an effect that is opposite but equal to (something) : to balance (something) by being opposite

Answer 4

Thanks to all for the suggestions - you have led me to a good answer so I thought I would post it up as a response to my own question.

I did some searching around these two suggestions:

• @Phil Sweet : inverse
• @Will Kunkel : symmetry

I found this:

• The definition of the inversion symmetry operator is that it transforms a vector into a different vector of same magnitude but antiparallel orientation.

( Nice to see the word antiparallel resurfacing )

So going back to my original question and examples - I'll go with this statement:

" The following can all be classed as inversion symmetry pairs "

NUMBER

• ( +5 ) <> ( -5 )

VECTOR / FORCE

• ( magnitude 5 angle 0 degrees ) <> ( magnitude 5 angle 180 degrees )

ELECTRIC CHARGE

• ( 2+ ) <> ( 2- )

I may refine the terms a bit more, but I'm pretty comfortable with that.