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

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. AB = BA

(ii) Scalar product is distributive,
i.e. A • (B+ C) = AB + AC

(iii) Scalar product of two perpendicular vectors is zero.
AB = 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.
AB = AB cos o° = AB (for parallel)
AB = 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.
AA = 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
$$\mathbf{A} \cdot \mathbf{B}=\left(A_{x} \hat{\mathbf{i}}+A_{y} \hat{\mathbf{j}}+A_{z} \hat{\mathbf{k}}\right) \cdot\left(B_{x} \hat{\mathbf{i}}+B_{y} \hat{\mathbf{j}}+B_{z} \hat{\mathbf{k}}\right)$$
= AxBx + AyBy + AzBz

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