Given the vectors \(\overrightarrow{u}\) and \(\overrightarrow{v}\), the real number obtained from the operation \(\overrightarrow{u}\cdot \overrightarrow{v}\) such that \(\overrightarrow{u}\cdot \overrightarrow{v} = \left | \overrightarrow{u} \right | \cdot \left | \overrightarrow{v} \right |\cos\theta \), where \(\left| \overrightarrow {u}\right|\) represents the norm of the vector \(u\), \(\left | \overrightarrow{v} \right |\) represents the norm of the vector \(v\) and \(\theta\) is the measure of the angle formed between the directions of the two vectors.

If the Cartesian components of the vectors \(\overrightarrow {u}\) and \(\overrightarrow {v}\) are \((a,b)\) et \((c,d)\), respectively, then \(\overrightarrow{u}\cdot \overrightarrow{v}=ac+bd\).

Therefore, the scalar product of two vectors is a real number (a *scalar*).

### Educational notes

- The scalar product is different from the multiplication of a vector by a scalar in that:
- The scalar product of two vectors is a real number; the two operands of a scalar product are vectors.
- The operands of the multiplication of a vector by a scalar are a vector and a real number; the result of the multiplication of a vector by a scalar is a vector.

- The expression “vector multiplication,” which should refer to an internal operation on the set of vectors and have a vector as a result, is inappropriate, as the
*scalar product*of two vectors is a real number and not a vector, whereas the multiplication of a vector by a scalar is an external operation.