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There are many physical quantities like weight, mass, force, velocity, and pressure which come in our contact. Some of these quantities have magnitude, and some have magnitude as well as direction.
These physical quantities can be broadly categorized into two types of quantities which are known as scalar and vector. These quantities give us information about the direction and magnitude. Let us understand in detail the scalar and vector quantities.
Scalar Quantities
There are some quantities that deal with numerical values like temperature, pressure, speed, etc. These quantities are described as scalar quantities. In other words, the physical quantities which have only magnitude are described as scalar quantities. These quantities do not have directions but magnitude only.
Properties of Scalar Quantities
- Commutative – Scalar quantities when multiplied are commutative. That is, A x B= B x A.
- Distributive – Scalar quantities are distributive over addition. That is, A (B + C) = AB + AC.
- Associative – Scalar quantities follow the associative property. A(BC) = (AB)C.
Operations on Scalar Quantities
In scalar quantities, the operations use simple algebra rules of addition, subtraction, and multiplication.
The results of these operations are always scalar.
Vector Quantities
Vector quantities are those quantities that have magnitude as well as directions. These are denoted by a. The quantities like displacement, force, electric field, and momentum are some examples of vector quantities. Those vector quantities having a unit magnitude are known as unit vectors and are represented by a and read as ‘a hat’ or ‘a cap’.
Operations on Vector Quantities
In vector quantities, we cannot use rules of the algebra. Some special rules are used to operate vector quantities. Let us understand the operations that can be performed on vector quantities.
- Addition of vector – Vector addition can be done by using the triangle law addition or parallelogram law addition. The addition of vectors follows associative and commutative properties.
- Scalar multiplication – When a vector is multiplied with a scalar quantity, its magnitude changes; however, the direction remains the same. It can be written as
Kp = k|p|
Here, |p| represents the magnitude of vector p .
- Multiplication of two vectors – The multiplication of two vectors is divided into two categories, one is the dot product and the other is the cross product. The dot product is also known as a scalar product since it always gives us the scalar value, whereas the cross product is also known as a vector product because the result of the vector product is always a vector. The dot product of two vectors, say a and b, is represented as a . b and the cross product is represented as a x b .
Properties of Vector Quantities
- The addition of vectors is commutative and associative in nature.
- Vectors are distributive over addition in scalar and vector forms which means k (b + c) = ka + kc and in vector form a (b + c) = a . b + a .c.
- For every vector, a , there is a vector -a which gives a +- a = 0.
Difference Between Scalar and Vector Quantities
S.NO | Scalar Quantity | Vector Quantity |
1. | Scalar quantities are always one-dimensional | Vector quantities can be dimensional or two-dimensional. |
2. | These quantities are related to the magnitude only. | These quantities are related to magnitude as well as direction. |
3. | The scalar quantities can be added or subtracted using simple algebraic rules. | For the addition of vectors, quantities can be done by using the triangle law of vector addition or parallelogram of vector addition. |
4. | We can perform division operations on a scalar quantity. | We cannot divide two vector quantities. |
5. | Some examples are length, mass, volume, time, speed, and density | Some examples are momentum, velocity, displacement, force, and angular velocity |
Applications of Vectors and Scalars
In the field of mathematics and physics, vectors and scalars are very useful. Calculating the speed of any ball, the distance covered by a train or a vehicle, the mass of any object, and calculating the temperature scalar quantities plays a very important role in our lives. Vector quantities are also very useful in the fields of engineering and physics for calculating electric fields, magnetic fields, gravitational force, and flow of fluid. Apart from this, these quantities are very useful in biological and medical applications.
Conclusion
We have been introduced to scalars and vectors, which are the quantities that help us to understand the direction and magnitude of any physical quantity. The scalar quantities are always represented in the form of numerical values only. The operations on scalar quantities can be performed by using simple algebra whereas operations on vector quantities cannot be done by using basic algebra rules. We have also learned about the difference between scalar and vector quantities and real-life applications of scalars and vectors.
