# Trouble with the definitions of equivalent and collinear.

1. May 14, 2013

My teacher, textbook and the internet have differing definitions.

First of all: Equivalent.

My teacher says that two parallel vectors with the same magnitude are equivalent, but my textbook says that two vectors in the same direction are equivalent.

×-->
<--×

are these equivalent?

And Collinear.

I can't find any lesson on collinear vectors online. All the definitions say two points are collinear when they lie on the same line, but nothing about vectors.

My textbook says "Vectors are collinear if they can be translated so that their start and end points lie on the same line"

But my teacher says two vectors are collinear if one is a scalar multiple of the other

So first, to clarify, magnitude has no impact on whether two vectors are collinear?

And second, the textbook definition seems to be describing parallel vectors.

×-->
×----->

Are these collinear? They can be translated to be on the same line.

<---×--->

Are these collinear? They are scalar multiples (negative 1) of each other.

<---× ×--->

2. May 14, 2013

### haruspex

I'm not familiar with the term 'equivalent vector'.
The confusion arises because a vector does not have a 'line of action'. It has a magnitude and an orientation only. A force has a line of action, so is more than just its vector representation.
I do see references to 'localized vectors', meaning that the line of action is also specified.

It doesn't mean anything to 'translate' a vector unless you mean localized vectors. You can translate a force by changing its line of action. If you ignore the reference to translation, the two definitions of collinear are the same.

Similarly, if two vectors have the same magnitude and direction they are the same vector, whereas if we're discussing localized vectors then they are merely equivalent, i.e., correspond to the same non-localized vector.