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I am a bit confused with the concept of vectors. If I have understood correctly from a linear algebra perspective, vectors are just elements of a vector space (which is a set V along with addition on V and scalar multiplication on V such that some properties hold). In this sense, vectors can be translated so that the vector from (0,0) to (1,2) is the same vector as the one that takes from (42,1000) to (43,1002). Where I get confused is when I am trying to view things from the "physics" perspective where the concept of bound (or located or position) vectors appear. For example, the linear velocity is usually defined as v = r x ω, where ω is the rotational velocity vector of a body and r is the position vector. I am just wondering how is it possible to call r a vector when in the case of arbitrary translation of this vector the velocity results would change?

I have seen a relevant answer but I do not understand what the answer means with "attaching" a vector to a point. (https://math.stackexchange.com/a/2589979/1004963)

urovorros
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  • What do you mean the velocity results would change ? It must be remembered that angular velocity is measured about a point, so if you change that point the angular velocity will change. – Rene Schipperus Jan 13 '22 at 14:43
  • Using vectors in physics is like using numbers in accounting. You place them where you need them. For example, the vector $(1,2)$ can represent a velocity that we draw tangent to a curve at a point.

    A vector space is like the number line. There is only one copy of each element and it doesn't move. So the vector $(1,2)$ in the vector space $\mathbb R^2$ occupies a unique position but that vector can be used in many applications.

     – John Douma Jan 13 '22 at 15:29
  • Thank you, guys! That cleared my missunderstanding! I appreciate it! Happy new year! – urovorros Jan 13 '22 at 15:51

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For example, the linear velocity is usually defined as $\mathbf v = \mathbf r \times \mathbf ω,$ where $\mathbf ω$ is the rotational velocity vector of a body and $\mathbf r$ is the position vector. I am just wondering how is it possible to call $\mathbf r$ a vector when in the case of arbitrary translation of this vector the velocity results would change?

Porting the position vector $\mathbf r$ to a different location maintains both its direction and magnitude, and also doesn't affect the angular velocity $\mathbf ω,$ which continues to point perpendicular to the plane on which $\mathbf r$ lies. Consequently, the linear velocity $\mathbf v$ is unchanged, even if we visualise it as too being (correspondingly) ported together with $\mathbf r.$

ryang
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