A vector admits multiples and submultiples.

Given the vector **v**, its m-nth multiple **w** is the sum of m vectors
each equal to **v**; we can write
**w** = m**v**;

conversely, there exists a vector **u** such that **v** = n**u**;
**u** is the n-nth submultiple of **v** and we can write

Multiples and submultiples of **v** have the same direction as **v**.

**y**, the m-nth multiple of **u** can be written as
;
it too has the same direction as **v** but its length is
with respect to that of **v**.

Therefore we can define the product of a *rational number* times a vector
and, more generally, the product of a *real number* times a vector; in this
context, the real number is said a **scalar**.

Scalar times a vector is a vector with same direction as the multiplicand vector and magnitude equal to its magnitude times the scalar.

If we note
the magnitude of **v**, we have that
is a vector with the same direction as **v** and magnitude = 1.
When a vector has magnitude=1, it is said a **unit vector**; often unit vectors are
called **versors**.

The unit vector
is
therefore said the **versor** of **v**.

Given two vectors **a** and **b** applied to the same point O,
let α be their plane. If the the least rotation required to superimpose **a**
to **b** is counterclockwise, we can represent such rotation as a versor
perpendicular to α and oriented from α to the observer:
this is the **rotation versor**.

If the rotation is clockwise, the rotation versor is opposite to the former.

Two vectors **a** and **b** applied to the same point O determine
a parallelogram with sides equal to the magnitudes of the vectors.
If we multiply the rotation versor of **a** and **b** by the area of this parallelogram,
we obtain a vector **c** said the **cross product** of **a** and **b**.

The cross product **c** is written as

and sometimes as

The **cross product is not commutative**. If we change the vectors' order
we obtain the opposite vector:

**The cross product of parallel vectors is null**.

In particular, **the cross product of a vector by itself is null**.

The magnitude of the cross product of two given vectors has a maximum when the vectors are perpendicular.

The area of a parallelogram coincides with the area of a rectangle with
equal base and equal altitude. If we know the lengths *a* and *b*
of its sides and an angle *θ* between them and let *a* be the base,
the altitude of the rectangle must be *b sin θ*. The magnitude of the cross product
is therefore

We can also associate to two given vectors **a** and **b**, applied to the same
point O, the area of the rectangle in which a side has length given by the magnitude of either
vector, say **a**, and the other side has length equal to that of the projection
of the second vector, say **b**, on the line of action of the first vector.
Furthermore, we can consider this area as negative if the projection lies outside the
first vector, positive (or eventually null) otherwise.

The **real number** so obtained is said the **dot product** of **a** and **b**.

It must be emphasized that **the dot product is not a vector**, but a **real number**.

The dot product **is commutative**.

If two vectors are perpendicular, their dot product is null.

Given two vectors, the magnitude of their dot product has a maximum when they are parallel.

The square of a vector is given by the square of its magnitude.

If we know the magnitudes of **a** and **b** and the angle *θ*
between them, the length of the projection of *b* on the line of action of **a** is
given by *b cosθ*, so the dot product is