Scalar Product (Dot Product)

The scalar product, or dot product, of two vectors equals the product of their magnitudes multiplied by the cosine of the angle between them. This operation yields a scalar quantity, not a vector. The scalar product can also be computed as the projection of one vector onto another, multiplied by the magnitude of that other vector.

Work is defined as the scalar product of force and displacement vectors. When force acts horizontally, it contributes fully to work. When force acts at an angle, only the component in the displacement direction contributes. A vertical force produces zero work during horizontal displacement because the angle between them is 90 degrees.

The scalar product is commutative and distributive. The dot product of orthogonal unit vectors equals zero, while the dot product of a unit vector with itself equals one. Parallel vectors have a dot product equal to the product of their magnitudes, and antiparallel vectors also yield the product of their magnitudes.

The angle between two vectors is found by taking the inverse cosine of the ratio of their dot product to the product of their magnitudes. This formula rearranges the definition of scalar product to solve for the angle, allowing you to determine the angular separation between any two vectors.

The dot product of two vectors equals the sum of the products of their corresponding components. Using vector components in the Cartesian coordinate system, you multiply each pair of matching components and add the results. This component method provides a practical computational approach without needing to know the angle between vectors.

The dot product of a vector with itself equals the square of its magnitude. Since the angle between a vector and itself is zero degrees, and cosine of zero is one, the result is the magnitude multiplied by itself, yielding a positive scalar quantity.

The scalar product is called a scalar because it produces a single numerical value, not a vector with direction. Unlike vector operations that result in directional quantities, the dot product of two vectors always yields a magnitude-only result, which is the defining characteristic of a scalar quantity.