2.2 Acceleration and deceleration  (Page 2/4)

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

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 “-5 i " 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.

“increase” and “decrease” of vectors quantities

The vector consist of both magnitude and direction. There can be infinite directions of a vector. On the other hand, increase and decrease are bi-directional and opposite concepts. Can we attach meaning to a phrase “increase in direction” or “decrease in direction”? There is no sense in saying that direction of the moving particle has increased or decreased. In the nutshell, we can associate the concepts of increase and decrease with quantities which are scalar – not quantities which are vector. Clearly, we can attach the sense of increase or decrease with the magnitudes of velocity or acceleration, but not with velocity and acceleration.

For this reason, we may recall that velocity is defined as the time rate of “change” – not “increase or decrease” in displacement. Similarly, acceleration is defined as time rate of change of velocity – not “increase or decrease” in velocity. It is so because the term “change” conveys the meaning of “change” in direction as well that of “change” as increase or decrease in the magnitude of a vector.

However, we see that phrases like “increase or decrease in velocity” or “increase or decrease in acceleration” are used frequently. We should be aware that these references are correct only in very specific context of motion. If motion is unidirectional, then the vector quantities associated with motion is treated as either positive or negative scalar according as it is measured in the reference direction or opposite to it. Even in this situation, we can not associate concepts of increase and decrease to vector quantities. For example, how would we interpret two particles moving in negative x-direction with negative accelerations $\begin{array}{l}-10m/s^2\end{array}$ and $\begin{array}{l}-20m/s^2\end{array}$ respectively ? Which of the two accelerations is greater acceleration ? Algebraically, “-10” is greater than “-20”. But, we know that second particle is moving with higher rate of change in velocity. The second particle is accelerating at a higher rate than first particle. Negative sign only indicates that particle is moving in a direction opposite to a reference direction.