A vector like acceleration may be directed in any direction in three dimensional space, which is defined by the reference coordinate system. Now, why should we call vector (as shown in the figure below) represented by line (i) positive and that by line (ii) negative? What is the qualification for a vector being positive or negative? There is none. Hence, in pure mathematical sense, a negative vector can not be identified by itself.

Vector representation in three dimensional reference |
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In terms of component vectors, let us represent two accelerations

Why should we call one of the above vectors as positive and other as negative acceleration? Which sign should be considered to identify a positive or negative vector? Further, the negative of

So, what is the actual position? The concept of negative vector is essentially a relative concept. If we represent a vector A as shown in the figure below, then negative to this vector –A is just another vector, which is directed in the opposite direction to that of vector A and has equal magnitude as that of A.

In addition, if we denote “-A” as “B”, then “A” is “-B”. We, thus, completely loose the significance of a negative vector when we consider it in isolation. We can call the same vector either “A” or “-A”. We conclude, therefore, that a negative vector assumes meaning only in relation with another vector.

Vector representation in three dimensional reference |
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In one dimensional motion, however, it is possible to assign distinct negative values. In this case, there are only two directions; one of which is in the direction of reference axis (positive) and the other is in the opposite direction (negative). The significance of negative vector in one dimensional motion is limited to relative orientation with respect to reference direction. In the nutshell, sign of vector quantity in one dimensional motion represents the directional property of vector. It has only this meaning. We can not attach any other meaning for negating a vector quantity.

It is important to note that the sequence in “-5i" is misleading in the sense that a vector quantity can not have negative magnitude. The negative sign, as a matter of fact, is meant for unit vector “i”. The correct reading sequence would be “5 x -“i”, meaning that it has a magnitude of “5” and is directed in “-“i” direction i.e. opposite to reference direction. Also, since we are free to choose our reference, the previously assigned “-5 i” can always be changed to “5 i” and vice-versa.

We summarize the discussion so far as :

- There is no independent meaning of a negative vector attribute.
- In general, a negative vector attribute is defined with respect to another vector attribute having equal magnitude, but opposite direction.
- In the case of one dimensional motion, the sign represents direction with respect to reference direction.