**Scalar Product of Two Vectors a and b is:**

The scalar product of two vectors is equal to the product of their magnitudes and the cosine of the smaller angle between them. It is denoted by (dot).

A • B = AB cosθ

The scalar or dot product of two vectors is a scalar.

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## Scalar Product of Two Vectors Definition in Physics – Scalars and Vectors

**Scalar or Dot Product Properties**

(i) Scalar product is commutative,

i.e. **A** • **B** = **B** • **A
**

(ii) Scalar product is distributive,

i.e.

**A**• (

**B**+

**C**) =

**A**•

**B**+

**A**•

**C**

(iii) Scalar product of two perpendicular vectors is zero.

**A**•

**B**= AB cos 9o° = 0

(iv) Scalar product of two parallel vectors or anti-parallel vectors is equal to the product of their magnitudes, i.e.

**A** • **B** = AB cos o° = AB (for parallel)

**A** • **B** = AB cos 180° = -AB (for anti-parallel)

(v) Scalar product of a vector with itself is equal to the square of its magnitude, i.e.

**A** • **A** = AA cos 0° = A²

(vi) Scalar product of orthogonal unit vectors

\(\begin{array}{l}

\hat{\mathbf{i}} \cdot \hat{\mathbf{i}}=\hat{\mathbf{j}} \cdot \hat{\mathbf{j}}=\hat{\mathbf{k}} \cdot \hat{\mathbf{k}}=1 \\

\hat{\mathbf{i}} \cdot \hat{\mathbf{j}}=\hat{\mathbf{j}} \cdot \hat{\mathbf{k}}=\hat{\mathbf{k}} \cdot \hat{\mathbf{i}}=0

\end{array}\)

(vii) Scalar product in cartesian coordinates

\(\mathbf{A} \cdot \mathbf{B}=\left(A_{x} \hat{\mathbf{i}}+A_{y} \hat{\mathbf{j}}+A_{z} \hat{\mathbf{k}}\right) \cdot\left(B_{x} \hat{\mathbf{i}}+B_{y} \hat{\mathbf{j}}+B_{z} \hat{\mathbf{k}}\right)\)

= A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z
}

**Scalars and Vectors Topics:**