The scalar Product: Product Of Two Vectors

Many physical quantities we deal with are represented as vector quantities, such as velocity, force etc. These quantities interact with each other to produce a resultant effect. In order to find the resultant of these forces, operations such as addition, subtraction and multiplication are required to be performed by these forces. In this section, we will learn about the multiplication of two vector quantities.

There are two types of multiplication of vectors possible; the scalar multiplication, which produces a scalar as the product of the multiplication and the other is vector multiplication, which produces a vector as a product. Here we will learn about the scalar product of two vectors.

Scalar Product

Let us consider two vectors A and B. The dot product of these two vectors is given as

vectors A and B

, Where is the angle between these two vectors?

The scalar product can also be written as,

vectors A and B
Scalar product

As we know BcosƟ is the projection of B onto A and AcosƟ is the projection of A on B, the scalar product can be defined as the product of the magnitude of A and the component of B along with A or the product of the magnitude of A and the component of B along with A.

Commutative law

Commutative law

Distributive Law

Distributive Law

Where λ is a real number.

Let us discuss the dot product of two vectors in a three-dimensional motion. Consider two vectors represented in terms of three unit vectors,

dot product

Where, is the unit vector along the x direction,  is the unit vector along the y direction and is the unit vector along the z direction.

The scalar product of the two vectors is given by,

scalar product

Here,

scalar product
scalar product

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