# Multiplying a vector by a number

Representation of a vector AB

In our studies of vector quantities, we make use of an arrow whose direction always points to the right. This arrow is called a vector and, by definition, is a mathematical entity that represents the set of oriented line segments that have a module, a direction and a direction. In several situations we can use vectors, either in addition, subtraction or multiplication. When multiplying two vectors, we must make the product of the vector by its numerical value. Below is the definition of vector multiplication.

Let’s imagine a real number whose value is k, where k ≠ 0 and a vector . The product of k by is a vector , represented by:

If k > 0, and have the same sense;

If k < 0, and have opposite directions;

If k = 0 or , the product is the zero vector.

Since k ≠ 0, the product can be denoted by .

For example, in the figure below we have a vector with | | = ** U** . Vector

**2**has magnitude 2 and the same direction as . The vector

**has magnitude 3 and the opposite direction to**

^{_}3In the figure above we have:

| **2 | = |2| . | ****| = 2 u**

**|**| = |

^{_}3**| =**

^{_}3| . |**3**

*u*