Archived May, 2026.

Associativity of vector addition

mark

Recall that the addition of real numbers is known be both commutative and associative. That is, if a, b, and c are real numbers, then

  • a+b = b+a (commutativity) and
  • (a+b)+c = a+(b+c).

Similarly, we proved in class that the addition of two-dimensional vectors is commutative. That is, we showed that if \vec{u} and \vec{v} are two-dimensional vectors, then

\vec{u}+\vec{v} = \vec{v} + \vec{u}.

You might also recall that we used a so-called componentwise proof to justified this and commented that it should extend naturally to higher dimensional vectors.

Questions

  1. How would you define the associativity of vector addition?
  2. How would you prove the associativity of vector addition?
User 001

I would assume vector addition would be described the same, (\vec{v} + \vec{u}) + \vec{w} = \vec{v} + (\vec{u} + \vec{w}).

Proof:

Let \vec{u} = \langle u_1,u_2\rangle, \vec{v} = \langle v_1,v_2\rangle, and
\vec{w} = \langle w_1,w_2\rangle be vectors in \mathbb{R}^2.
Using component-wise addition

(\vec{u}+\vec{v})+\vec{w} = \langle u_1+v_1,\, u_2+v_2\rangle + \langle w_1,w_2\rangle = \langle (u_1+v_1)+w_1,\; (u_2+v_2)+w_2\rangle.

Likewise,

\vec{u}+(\vec{v}+\vec{w}) = \langle u_1,u_2\rangle + \langle v_1+w_1,\, v_2+w_2\rangle = \langle u_1+(v_1+w_1),\; u_2+(v_2+w_2)\rangle.

Since the components are real numbers and addition for real numbers is associative,

(u_1+v_1)+w_1 = u_1+(v_1+w_1), \quad (u_2+v_2)+w_2 = u_2+(v_2+w_2).

Therefore,

(\vec{u}+\vec{v})+\vec{w} = \vec{u}+(\vec{v}+\vec{w}),

so vector addition is associative.