show that the scalar product of vector obeys the commutative law
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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 vector products 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 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