The set together with the energetic inner product is a pre-Hilbert space.

Equipped with this inner product, L is in fact complete.

This result is perhaps most transparent by considering the inner product defined above.

The relationship remains true independent of the frame in which the inner product is calculated.

The pairing between these two spaces also takes the form of an inner product.

This is slightly different than the above definition, which permits a change of inner product.

The inner product of two vectors is a complex number.

The inner product of the 4-acceleration and the 4-velocity is therefore always zero.

Two such vectors are adjacent when their inner product is 8.

We can calculate the value of n by considering the inner product.

These results are equivalent to the equation containing the dot product.

This property of the dot product has several useful applications (for instance, see next section).

The dot product of force and distance is mechanical work.

The opposite is true for the dot product of two unit vectors.

Where and agree, so those terms affect the dot products equally.

However, the rules for dot products do not turn out to be simple, as illustrated by:

The flux can be written as the dot product of the field and area vector.

Now to find intersection point with the clipping window we calculate value of dot product.

The reason for the dot product is as follows.

These results are equivalent to the dot product between velocity and the normal direction to the area.

The scalar product of a four-velocity and the corresponding four-acceleration is always 0.

This formula does not explicitly depend on the definition of the scalar product.

The prototypical example of a pseudoscalar is the scalar triple product.

The scalar product of the differential 4-position with itself is:

This scalar product of force and velocity is classified as instantaneous power.

The scalar products met in vector analysis are familiar examples of contraction.

(see above), the scalar product of d and m must be zero!

Another simple way to look at it as a scalar product of vectors in module 10.

The scalar triple product is invariant under rotation of the coordinate system.

This definition therefore depends on the definition of the scalar product.