Vectors and their Notations

For students of Physics, there would be plenty of instances where they would have acquainted themselves with terms like vectors, scalars and so on. Here the topic for discussion would be about vectors and their notations.

In most cases, it is necessary that a typical physical quantity has to be provided with a description and magnitude description and direction in a complete manner. While carrying out the subtraction and addition of these quantities physically, the possibility of following the algebra rules is ruled out by all means. In the later part of this chapter, we would look into the rules designed for the subtraction and addition of vectors. Simply put, these quantities physically together constitute what we refer to as vectors.

For instance, let us take the example of ‘Force’ as it has direction and magnitude (value in numerical form) and hence a vector quantity. If in case one provides you with some information like 5 N being the force exerted on a particular object without mentioning the direction of the force, the information would be deemed incomplete. Such is the importance of direction in the case of force as it would turn the information complete. Certain other examples of vector quantities include Linear Momentum, acceleration and velocity.
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How vectors are represented:  

Let us take the instance of a velocity vector in the direction of x-axis (positive) of 5 unit magnitude. One could represent this using a parallel line drawing on velocity and to show the direction, mark an arrow for the same.

Vector Notations:

To denote vectors, arrows are marked above the representative symbols for them. Examples include \(\vec{AB}\), \(\vec{BC}\) and so on. Single letters also would be suitable for this purpose like:

Velocity vector:\(\vec{v}\)

Force vector: \(\vec{F}\)

Linear momentum: \(\vec{p}\)

Acceleration vector: \(\vec{a}\)

General representation

Vector representation

k . r \(\vec{k}\cdot \vec{r}\)
\(E\times B\) \(\vec{E}\times \vec{B}\)
\(\bigtriangledown \times r\) \(\vec{\bigtriangledown} \times \vec{r}\)
v = 0 \(\vec{v}=\vec{0}\)
\(\hat{r}\) \(\hat{\vec{r}}=\hat{r}\)

Equality of vectors:

If the directions and magnitudes of two vectors are same, the vectors would be equal in nature. However one should remember that vectors of same physical quantity are put into comparison.  For example, it would be feasible to compare Force vector of 5 N in positive x-axis and velocity vector of 5 m/s in the positive x-axis.

From the above discussion, it is obvious that this topic has great significance for students of Physics and would continue to do so in the years to come.

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