The Scalar Product

(also known as the dot product or the inner product)

The component of the vector a in the direction of OP is easily calculated as equivalent to the length ON by

Considering the constant force F in the figure below, which acts through the point O. If this force is moved along the line OA, along the vector a, then we can calculate the work done by the force. (Remember: work done is the component of the force in the direction of movement multiplied by distance moved by the point of application.)

The force F is move the distance giving,

This is the scalar product of the two vectors F and a and can be seen as a geometric definition.

An equivalent component definition can be written.

The scalar product of two vectors and . Is defined as

In geometrical form

Where is the angle between the two vectors and .

The two definitions can be proved to be equivalent by the cosine rule for a triangle,

Which can be expanded to show that

Three important points about a scalar product:

  1. The scalar product of two vectors gives a number (a scalar)
  2. The scalar product is a product of vectors
    (it cannot be of two scalars nor a vector and a scalar)
  3. Use of the "dot" is essential to indicate that the calculation is a scalar product.