Multiplication of Vector with Scalar

Introduction to Vector Multiplication

Have you ever wondered what is a vector and what can be done with vectors? If so, the answers to all your questions regarding vectors can be fetched in this article. A physical quantity is that quantity that can be measured physically using a scientific device. However, quantities such as hunger, love, depression, anger etc cannot be characterized as physical quantities because they cannot be measured manually. A physical quantity that has only magnitude is called a scalar quantity. A scalar quantity is direction independent. A physical quantity that has both magnitude and direction is called a vector quantity. Scalars are represented by straight line segments without any arrow heads whereas vectors are represented by straight lines with an arrow head one of the end points indicating the direction of the vector.

Multiplication of Vectors

Vector multiplication rules is one of the easiest and most interesting concepts in Mathematics. Vector multiplication is finding the product of any two vectors either as a scalar or as a vector. Multiplying vectors can be done in two forms namely dot product and cross product. If a vector is multiplied by a scalar it means that the magnitude of a vector is multiplied by a number.

Multiplying Vectors with Scalars

Though vectors and scalars represent different varieties of physical quantities, at times it is necessary for both of them to interact. Addition of a scalar to a vector quantity is highly impossible because of their differences in dimensions. However, a vector quantity can be multiplied by a scalar. At the same time, the converse of this is not possible. i.e. A scalar can never be multiplied by a vector.

During the multiplication of vectors with scalars, the similar quantities are subjected to arithmetic multiplication. i.e. the magnitude of vectors is multiplied with that of the scalar quantities. The product obtained by multiplying vectors with scalars is a vector. The product vector has the direction same as that of the vector which is multiplied with the scalar and its magnitude is increased as much number of times as the product of the magnitudes of vector and scalar that are multiplied.

Scalar Vector Multiplication Rules Example

Scalar Vector Multiplication Example 1

Consider a certain vector say vector ‘a’ is multiplied with a scalar whose magnitude is 0.25. In this case, the product vector is a vector which represents a vector whose direction is the same as that of vector ‘a’ and the magnitude is equal to ¼ times that of the vector ‘a’ (because 0.25 represents ¼).

Scalar Vector Multiplication Example 2:

The physical quantity force is a vector quantity. The work done is dependent on both magnitude and direction in which the force is applied on the object. This force is actually a product of a vector with a scalar quantity as per Newton’s second law of linear motion. 

The force is given as:

F = m x a

In the above equation, ‘a’ denotes the acceleration which is a vector quantity and ‘m’ denotes the mass of the object which is scalar.

So, it is one of the examples in physics for the multiplication of vectors with scalars.


Scalar Vector Multiplication Example 3

Let any arithmetic number which is purely unitless be taken as the scalar quantity. On multiplying vectors with this scalar, the product obtained is a scaled version of the initial vector. Suppose the number considered as a scalar is 3, then the vector if multiplied by this scalar yields a product vector which is the same as three times the initial vector. 


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Practical Applications of Multiplication of Vectors with Scalars

Multiplication of vectors with scalars find a wide range of applications in Physics. Many SI units of the vector quantities are the products of the vector and scalars. For example, the SI unit of velocity is meter per second. Velocity is a vector quantity. This is obtained by multiplying the two scalar quantities: length and time with a unit vector in a specific direction. There are many other instances in Mathematics and Physics where the vector multiplication with scalar is used.

Fun facts About Vector Multiplication Rules

  • A vector can be multiplied by a scalar. But, a scalar quantity cannot be multiplied by a vector. 

  • When a vector is multiplied with a scalar, the product obtained is a vector with the same direction but increased magnitude.

FAQ (Frequently Asked Questions)

1. What are the different types of products obtained when vectors are multiplied with vectors?

When a vector is multiplied with a vector, different types of products can be obtained. They are:

  • Dot Product: It is also called a scalar product. It involves multiplication of vectors to give a scalar product. 

  • Cross Product: It is also called a vector product. It involves multiplying vectors to yield a vector product.

  • Hadamard Product: This involves a technique to calculate the vector products in a sequential manner.

  • Triple products: It is the product of three vectors.

  • Multiple Cross Products: It is the vector multiplication involving more than three vectors.

2. What are different types of vectors?

A vector is a physical quantity which has both magnitude and direction. It is represented by a straight line with an arrow head at one of the ends indicating the direction of the vector. The different types of vectors are:

  • Zero Vector

  • Unit vector

  • Position vector

  • Coinitial vector

  • Like and unlike vectors

  • Coplanar vectors

  • Collinear vectors

  • Equal vectors

  • Negative of a vector

  • Displacement vector