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
\(\mathbf{A} \cdot \mathbf{B}=\left(A_{x} \hat{\mathbf{i}}+A_{y} \hat{\mathbf{j}}+A_{z} \hat{\mathbf{k}}\right) \cdot\left(B_{x} \hat{\mathbf{i}}+B_{y} \hat{\mathbf{j}}+B_{z} \hat{\mathbf{k}}\right)\)
= AxBx + AyBy + AzBz
Scalars and Vectors Topics: