# Reason behind vector addition law

What is the reason behind triangle law of vector addition, in other words, how is this really justified?

• Because this is how displacements, velocities, accelerations, and forces are observed to combine. – G. Smith Jun 29 '19 at 18:02
• @ G. Smith. Yes indeed. Perhaps another way of asking the question would be: Is there a general theory that $predicts$ the triangle law (or, as I prefer to call it, the head-to-tail rule) for velocities, forces and other vectors. – Philip Wood Jun 30 '19 at 8:14
• Yes is there any such theory? – Arnav Mishra Jul 2 '19 at 3:50
• Would Mathematics be a better home for this question? – Qmechanic Jul 2 '19 at 15:11

## 3 Answers

A naïve answer...

If you go from A to B then from B to C, you can represent your displacement from A to B as an arrow and from B to C to another arrow. Clearly your displacement from A to C can be represented by an arrow going from A to C or by the two arrows already mentioned, placed with the tail of the second touching the head of the first. This is the rule for adding displacements and arguably it is self-evident. The rule can be extended to any number of displacements.

Velocity is displacement per unit time and so velocities must add as displacements.

Momentum and acceleration are defined in terms of velocity, so momenta and accelerations must add as displacements.

The argument can be extended, via Newton's second law, to forces and field strengths.

• The argument can be extended to forces and field strengths. I liked everything up to here, but this seems like a weak point in this answer. Maybe a better justification for generalizing from displacement vectors to all vectors is that any two vector spaces of the same finite dimension are isomorphic. – Ben Crowell Jun 29 '19 at 20:52
• Does that mean vectors be better visualised as displacement? – Arnav Mishra Jun 30 '19 at 7:14
• Better, I wonder, than what? If I add vectors by the head-to-tail rule, I suppose that I do visualise them as displacements, but visualisation is to do with the relationship between you and the Physics, and not really part of the Physics itself. Perhaps Bill Watts (see next answer) 'visualises' adding vectors as a process of adding components. – Philip Wood Jun 30 '19 at 7:54

Well I finally realized that we are actually not adding two quantities but trying to find the resultant. Therefore the resultant must indeed be the third side of the triangle as it will give us the shortest distance between the two points, i.e the tail of one and the head of another. By definition the resultant is that vector which can have the very same effect as the other two vectors would provide together.

• The process that you describe for finding the resultant of two vectors is called 𝑎𝑑𝑑𝑖𝑛𝑔 the two vectors! The resultant is the $sum$ of the vectors.– – Philip Wood Jul 3 '19 at 11:29
• Well that was a more theoretical approach – Arnav Mishra Jul 3 '19 at 14:21
• I don't know about "theoretical approach"; I'm simply stating how the words "add" and "sum" are used in relation to vectors. – Philip Wood Jul 3 '19 at 16:32

When you add two vectors, you form the resultant vector by adding the components of the two individual vectors. The triangle law of vector addition is a visual equivalent to that.

• This doesn't really answer the question. It just justifies one definition in terms of another definition. – Ben Crowell Jun 29 '19 at 20:48
• Actually it answers the question well. The OP is not asking for justification of adding components. Vector addition is adding components. That is just a plain fact. He is asking for the justification of the triangle law of vector addition. Drawing the vectors with a triangle to show the resultant is a totally different process from adding components. But the reason that it works and the justification for it is that it is mathematically equivalent to adding components. – Bill Watts Jun 30 '19 at 7:17