Defining Scalar and Vector Products
The scalar product (or dot product) and the vector product (or cross product) are two distinct ways to multiply two vectors. The primary difference lies in the nature of their result: a scalar product yields a single numerical value (a scalar), while a vector product yields a new vector quantity.
Scalar Product (Dot Product)
The scalar product of two vectors, A and B, is denoted as A · B. It is calculated by multiplying the magnitudes of the two vectors by the cosine of the angle between them. Mathematically, A · B = |A| |B| cos(θ). The result is always a scalar, representing the projection of one vector onto another, or a measure of how much two vectors point in the same direction.
Vector Product (Cross Product)
The vector product of two vectors, A and B, is denoted as A × B. It results in a new vector that is perpendicular to both original vectors. Its magnitude is calculated by multiplying the magnitudes of the two vectors by the sine of the angle between them: |A × B| = |A| |B| sin(θ). The direction of the resulting vector is determined by the right-hand rule.
Applications in Science and Engineering
Scalar products are crucial for calculating work done by a force (Force · Displacement) or electrical power (Voltage · Current for DC). Vector products are essential in physics for concepts like torque (Position vector × Force), magnetic force on a moving charge (Charge × (Velocity × Magnetic field)), and angular momentum (Position vector × Linear momentum), where the direction of the resulting quantity is critical.