scalar product
dot product

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

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

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

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!

(the dot represents the scalar product of the two vectors).

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

The standard scalar product defined on has the signature (n, 0, 0).

For this reason, not every scalar product space is a normed vector space.

This result follows from a nice integral formulation of the scalar product:

In his work, the concepts of linear independence and dimension, as well as scalar products are present.

The ":" symbol represents the scalar product between matrices.

Now let us calculate the scalar product .

The invariant length of is where is a more general form of a scalar product.

The scalar product is given by .

This means that the signature is an invariant for scalar products on isometric transformations.

A positive semi-definite scalar product has a signature (n, 0, m).

If any two vectors of triple scalar product are equal, then its value is zero:

A projection operator requires a scalar product.

The law of total expectation holds, since the projection cannot change the scalar product by the constant 1 belonging to the subspace.

Other related quantities, the scalar products of pairs of four-vectors, are also invariant.

From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any pair of arguments.

A vector space equipped with a scalar product is called an inner product space.

The Bombieri norm is defined in terms of the following scalar product:

The dot product of force and distance is mechanical work.

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

The reason for the dot product is as follows.

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

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

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

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

The magnitude also can be expressed using the dot product:

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

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

The dot product between two 8-vectors is readily defined, and can be used to calculate the metric.

The double dot product between two 2nd order tensors is a scalar.

Furthermore, we can swap the bits and without changing whether or not the dot products are equal.

The real part will be the negative of the dot product of the two vectors.

An inner product is a generalization of the dot product.

It uses vectors in combination with the vector dot product.

In Euclidean geometry, the dot product, length, and angle are related.

The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system.

These are exactly equivalent formulations, merely using different notation for the dot product.

If the dot product is zero, the two vectors are said to be orthogonal to each other.

In some cases, the inner product coincides with the dot product.

This causes similar problems with definition of angle (see dot product) as appeared above for distances.

(where denotes the vector dot product and which is taken over ).

There are two ways to define the double dot product, one must be careful when deciding which convention to use.

The dot product is often defined in one of two ways: algebraically or geometrically.