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Two mutually orthogonal unit vectors acting at a point $p$, produce a resultant, whereas the two orthogonal unit basis vectors at the origin do not, why?

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  • $\begingroup$ Why do you think you can’t add two orthogonal unit vectors “at the origin”? And what point is $\hat i+\hat j$ “at”? $\endgroup$
    – Ghoster
    Commented Feb 5, 2023 at 5:26
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    $\begingroup$ This question is based on a false premise. $\endgroup$
    – Ghoster
    Commented Feb 5, 2023 at 5:28
  • $\begingroup$ Related post by OP: physics.stackexchange.com/q/748699/2451 $\endgroup$
    – Qmechanic
    Commented Feb 5, 2023 at 5:35
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    $\begingroup$ @NeilLibertine Scalar products do not produce a resultant vector, just a number. Also, OP is asking about vector addition. $\endgroup$
    – joseph h
    Commented Feb 5, 2023 at 7:14
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    $\begingroup$ "Sum of a vector is scalar product" That is a non-sequitur. "otherwise try to add 3 unit east and 4 unit north, is resultant 7 or 5" This makes 5 units, pointing north east. I still have no idea what it is you're trying to say. How does it relate to my comment or to yours about dot products? $\endgroup$
    – joseph h
    Commented Feb 5, 2023 at 8:09

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Vectors don't act - they're objects, not operators. The action is being done to one vector or the other by the operation of vector concatenation (more commonly "vector addition", which is a perfectly good term since we can express vectors as matrices or algebraic sums, both of which are concatenated with addition, and vector addition follows all the commutative, associative, and identity properties of addition).

If you operate on a basis vector by doing vector addition with another basis vector you get a resultant that's the vector sum of the basis vectors, as you would expect.

Note 1: Vectors also don't have location, although they may describe properties of mathematical objects that do (e.g. a point mass).

Note 2: we often talk about forces acting on objects, which seems (since force is a vector quantity) like it contradicts what I said - but it's just linguistic. When a force acts on an object, what we mean mathematically is that we need a new mathematical object to describe the physical object that having a particular physical interaction whose mathematical counterpart is a force vector.

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  • $\begingroup$ so the simplest way to say it is that basis vectors are simply vectors which are not (yet) added to each other, whereas the other two vectors (referred to in the question), are. ok but, In the derivation of the relativistic lorentz transformations between two inertial frames, no matter particles are referred to, only the velocity of the frame ( x', y',z', t' ) relative to the frame ( x, y, z, t ). If $ \vec{y} $ and $ \vec{x} $ are velocity vectors, wouldn't they be in different inertial frames? since dx/dy of $ \vec{y} $=0 whereas dx/dy of $ \vec{x} $=1 $\endgroup$
    – pete
    Commented Feb 5, 2023 at 7:48
  • $\begingroup$ @pete I'm not sure what you're asking, but it sounds like you have some kind of question about transforming between frames in relativity, which should be its own separate question, not a comment on an answer to a question about adding basis vectors. $\endgroup$
    – g s
    Commented Feb 5, 2023 at 8:07

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