Vectors can be multiplied using two primary methods: the dot product (also known as the scalar product) and the cross product (or vector product). The choice of method depends on the specific outcome you wish to achieve.

In physics, vectors are defined as quantities that possess both magnitude and direction. When performing vector multiplication, the two methods available serve different purposes.

The dot product is employed when the goal is to determine the magnitude of one vector in the direction of another. The dot product of two vectors, $\mathbf{A}$ and $\mathbf{B}$, is calculated using the formula:

where $|\mathbf{A}|$ and $|\mathbf{B}|$ represent the magnitudes of the vectors, and $\theta$ is the angle between them. The result of this operation is a scalar quantity, which means it has magnitude but lacks direction.

Conversely, the cross product is utilized when the objective is to find a vector that is perpendicular to the plane formed by the two original vectors. The cross product of $\mathbf{A}$ and $\mathbf{B}$ is given by the formula:

where $\mathbf{n}$ is a unit vector that is perpendicular to the plane containing $\mathbf{A}$ and $\mathbf{B}$. The direction of $\mathbf{n}$ is determined by the right-hand rule. The result of the cross product is a vector quantity, which means it has both magnitude and direction.

It is essential to recognize that vector multiplication is not commutative. Specifically, while the dot product satisfies the property $\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$, the cross product follows the rule $\mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A}$. This distinction is a fundamental aspect of vector operations and is crucial to remember when working with vectors.

In summary, when multiplying vectors, you can choose between the dot product and the cross product based on whether you seek a scalar or a vector result. The dot product provides the magnitude of one vector in the direction of another, while the cross product yields a vector that is perpendicular to the plane defined by the two original vectors.