Vector product or cross product of two vectors | Part-1

Vector Product or Cross Product of Two Vectors

The vector product, also known as the cross product, of two vectors results in a third vector that is perpendicular to both of the original vectors. This operation is defined in three-dimensional space and is denoted by \( \vec{A} \times \vec{B} \).

Definition:

Given two vectors \( \vec{A} = A_1\hat{i} + A_2\hat{j} + A_3\hat{k} \) and \( \vec{B} = B_1\hat{i} + B_2\hat{j} + B_3\hat{k} \), their cross product is defined as:

This determinant expands to:

Geometrical Meaning:

The magnitude of the cross product represents the area of the parallelogram formed by vectors \( \vec{A} \) and \( \vec{B} \). Mathematically:

\[ |\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta \]

Here, \( \theta \) represents the angle formed between the vectors \( \vec{A} \) and \( \vec{B} \).

Properties of Cross Product:

• Not Commutative: The order of multiplication matters, as \( \vec{A} \times \vec{B} = - (\vec{B} \times \vec{A}) \).
• Distributive over addition: \( \vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C} \)
• Scalar Multiplication: \( (k\vec{A}) \times \vec{B} = k(\vec{A} \times \vec{B}) \)
• Zero Vector: If vectors are parallel or anti-parallel, \( \vec{A} \times \vec{B} = \vec{0} \)
• Right-Hand Rule: Direction of \( \vec{A} \times \vec{B} \) follows the right-hand rule

Note:

The outcome of a cross product is a vector quantity, never a scalar.