# Scalar Product of Two Vectors

## Scalar Product of Two Vectors

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

1. 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.
2. 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.