2.8: Scalar Product (Dot Product)
The scalar multiplication of two vectors is known as the scalar or dot product. As the name indicates, the scalar product of two vectors results in a number, that is, a scalar quantity. Scalar products are used to define work and energy relations. For example, the work that a force (a vector) performs on an object while causing its displacement (a vector) is defined as a scalar product of the force vector with the displacement vector.
The scalar product of two vectors is obtained by multiplying their magnitudes with the cosine of the angle between them. In the definition of the dot product, the direction of the angle between the two vectors does not matter and can be measured from either of the two vectors. The scalar product of orthogonal vectors vanishes. Moreover, the dot product of two parallel vectors is the product of their magnitudes, and likewise, the dot product of two antiparallel vectors is also the product of their magnitudes. The scalar product of a vector with itself is the square of its magnitude.
In the Cartesian coordinate system, scalar products of the unit vector of an axis with other unit vectors of axes always vanish because these unit vectors are orthogonal. The scalar multiplication of two vectors is commutative and obeys distributive law. The scalar product of two different unit vectors of axes is zero, and the scalar product of unit vectors with themselves is one. The scalar product of two vectors is used to find the angle between the vectors.
This text is adapted from Openstax, University Physics Volume 1, Section 2.4: Products of Vectors.