**Scalar Product of Two Vectors**

The physical quantities may be broadly classified into the vectors and the scalars. The quantities with magnitude and direction both are known as vector quantities while, the quantities with only magnitude are called scalar quantities.

**What is a Scalar Product?**

Scalar product is a special technique of multiplying two vectors. As the name suggests, we get a scalar quantity after multiplication. Scalar product is also known as dot product (since it is denoted by a dot between the vectors) or inner product.

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

\(\widehat{A}.\widehat{B}\,=\,AB\,\cos \theta \).

Where

θ = Angle between two vectors.

The scalar product can also be written as,

\(\widehat{A}.\widehat{B}\,=\,AB\,\cos \theta \,=\,A(B\,\cos \theta )\,=\,B(A\,\cos \theta )\).

As we know B cosθ is the projection of B onto A and A cosθ 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:**

\(\widehat{A}.\widehat{B}\,=\,\widehat{B}.\widehat{A}\).

**Distributive Law:**

\(\widehat{A}\lambda \widehat{B}\,=\,\lambda (\widehat{B}.\widehat{A})\).

Where,

λ = Real Number

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

\(\widehat{A}\,=\,{{A}_{x}}\widehat{i}\,+\,{{A}_{y}}\widehat{j}\,+\,{{A}_{Z}}\widehat{k}\),

\(\widehat{B}\,=\,{{B}_{x}}\widehat{i}\,+\,{{B}_{y}}\widehat{j}\,+\,{{B}_{z}}\widehat{k}\),

Where,

\(\widehat{i}\) = Unit vector along the x direction.

\(\widehat{j}\) = Unit vector along the y direction.

\(\widehat{k}\) = Unit vector along the z direction.

The scalar product of the two vectors is given by,

\(\widehat{A}.\widehat{B}\,=\,({{A}_{x}}\widehat{i}+{{A}_{y}}\widehat{j}+{{A}_{z}}\widehat{k}).({{B}_{x}}\widehat{i}+{{B}_{y}}\widehat{j}+{{B}_{z}}\widehat{k})\),

\(\widehat{A}.\widehat{B}\,=\,{{A}_{x}}{{B}_{x}}+{{A}_{y}}{{B}_{y}}+{{A}_{z}}{{B}_{z}}\),

Here,

\(\widehat{i}.\widehat{i}\,=\,\widehat{j}.\widehat{j}\,=\,\widehat{k}.\widehat{k}\,=\,1\),

\(\widehat{i}.\widehat{j}\,=\,\widehat{j}.\widehat{k}\,=\,\widehat{k}.\widehat{i}\,=\,0\).