Scalar Product of Two Vectors
Given the vectors [latex]\overrightarrow{u}[/latex] and [latex]\overrightarrow{v}[/latex], the real number obtained from the operation [latex]\overrightarrow{u}\cdot \overrightarrow{v}[/latex] such that [latex]\overrightarrow{u}\cdot \overrightarrow{v} = \left | \overrightarrow{u} \right | \cdot \left | \overrightarrow{v} \right |\cos\theta [/latex], where [latex]\left| \overrightarrow {u}\right|[/latex] represents the norm of the vector [latex]u[/latex], [latex]\left | \overrightarrow{v} \right |[/latex] represents the norm of the vector [latex]v[/latex] and [latex]\theta[/latex] is the measure of the angle formed between the directions of the two vectors.
If the Cartesian components of the vectors [latex]\overrightarrow {u}[/latex] and [latex]\overrightarrow {v}[/latex] are [latex](a,b)[/latex] et [latex](c,d)[/latex], respectively, then [latex]\overrightarrow{u}\cdot \overrightarrow{v}=ac+bd[/latex].
Therefore, the scalar product of two vectors is a real number (a scalar).
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.