Atlearner-
April 19, 2023

October 03, 2021

We know that anything that can be measured is called a physical quantity. And those that cannot be measured are not a physical quantity, their quality is observed only by observation through the senses such as taste, smell, sight, etc. But when it comes to physical quantities, different methods and mathematics are used in physics to measure them.

For example, there are some physical quantities such as mass, speed, velocity, force, and work, and so on. These quantities are often described as scalar or vector quantities.

So let us know what these scalar and vector quantities are. After that, we will learn how these quantities are measured?

Many physical quantities are such that they can be described just by their numerical value without any directions. We can add, subtract and multiply these physical quantities by using the simple rules of algebra. Where only their magnitudes are added, subtracted, and multiplied.

There are many scalar quantities, some of the common examples are - Mass, Speed, Distance, Time, Area, Volume, Density, Temperature, etc.

Many physical quantities are such that they are incomplete if we describe them with just their magnitude, they need direction to be completely described.

For example - Force is a vector quantity. So in order to describe the force, with the magnitude, it is also important to be mentioned in which direction the force is applied.

We can't add, subtract and multiply vectors by using the simple rules of algebra. Vector can be added, subtracted, and multiplied by vector addition, vector subtraction, and vector multiplication methods.

A vector quantity is represented by an arrow placed over the symbol of the given physical quantity. In general, any vector (say A) can be written as

Here ｜A｜ is the magnitude of the vector and n with cap (called a unit vector) shows the direction of the physical quantity A.

Note:

In the *3D* cartesian coordinate system to show the 3 dimensions, we use 3 unit vectors *i*,* j*, and *k* with a cap. To know more about the cartesian coordinate system read more...

There are many vector quantities, some of the common examples are - Displacement, Velocity, Momentum, Force, Work, Acceleration, Electric field, Magnetic field, etc.

Whatever we have learned from the above discussion is summarized below with some differences between scalar and vector for a better understanding.

Scalar quantity |
Vector quantity |

1. It is a
physical quantity that has only magnitude and no direction. |
1. It is a physical quantity that has both magnitude and direction. |

2. It is
represented with a number (magnitude) and Unit. |
2. It is represented with a number (magnitude), Unit, and direction using a unit cap or arrow at the top. |

3. A scalar
quantity is written as a quantity symbol. |
3. Vector quantity is written as in bold later or an arrow sign above the later. |

4. It has no directions. |
4. It has directions. |

5. Example -
Mass and Temperature |
5. Example - Velocity and Acceleration |

Vector addition and subtraction can be shows by using graphical and analytical methods. Although the graphical method of adding vectors helps us visualize vectors and the resultant vector but sometimes it is very tedious and gives limited accuracy.

It is much easier to add vectors by combining their respective component. So we can add and subtract vectors more accurately using analytical methods.

In the graphical method, vector addition can be performed using any of these two laws- triangle law and parallelogram law.

The resultant vector of two vectors A and B is given by,

If vector A makes an angle 𝜃 with B then the resultant vector R has magnitude,

This formula is well known as the law of cosines. Similarly, another formula known as the law of sines is also very popular in solving many problems. The formula goes like this

Similarly, Vector subtraction can be done as the addition of inverted vectors. Know more...

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