# B Mistake when explaining associativity of vector addition

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1. Jul 21, 2016

Does he make a mistake at 6:18?

In case of associativity. When doing a+(b+c) he's just placing the tip of a to the tail of (b+c), but shouldn't he have added the tail of a to the tip of (b+c) ending in a different point? I understand vector addition is associative, but I think how he did it is incorrect by accident. As far as I understand, this is how I see it:

2. Jul 21, 2016

### Staff: Mentor

He is right. He calculates $a + (b+c)$, i.e. $(b+c) = v$ first and then $a + v$. Your suggestion is $v + a$.
To add vectors, you have to apply them at the same point (and get a diagonal of a parallelogram) or concatenate them (and get the same diagonal). So even $v + a$ will lead to the same result, when applied to the origin.

The whole vector addition is already an abbreviation, since a vector should be $(P_v,v)$ where $P_v$ is the point the vector is applied to, i.e. where it starts. So
$$a + (b+c) = (P_a,a) + ((P_b,b) + (P_c, c)) = (P_a,a) + ((P_b,b) + (P_b +b, c)) = (P_a,a) + (P_b, b+c)) = (P_a, a+(b+c))$$
which is associative and commutative. So no matter how you write it, it's always the same result. Since parallel transport along straight lines doesn't change the resulting vectors, we drop the points where the vectors apply to in the notation for simplification.

3. Jul 21, 2016