The idea of **vector sum** appears when we perform an operation of **sum **vector. For example, if we use the **triangle method **to add two vectors ** A** y

**, we must draw them consecutively (respecting their modules, directions and directions), making the origin of**

*B***match end**

*B***; the vector adds**

*A***+**

*A***will have as its origin, the origin**

*B***and as an extreme the origin of**

*A***.**

*B***Definition**

Let there be two vectors ** A** y

**, he**

*B***vector sum**obtained by adding the vector

**with vector**

*A***That is, by adding the components of each vector:**

*B*** A** +

**= (A**

*B*_{x }+ B

_{x}, TO

_{y }+ B

_{y}, TO

_{z }+ B

_{z})

**Example: **Sean ** A **= (3, 2, -4) and

**= (-3, 2, 7), calculate the vector**

*B***+**

*A***.**

*B*** A** +

**= ( 3 + (-3), 2 + 2, -4 – 7) = (0, 4, 3)**

*B*### Sum of two vectors with the same direction and the same direction

Sean ** A** y

**:**

*B*To obtain the sum vector we follow the following steps:

- We draw the vector
below the vector*B*, so that they are consecutive, respecting their modules, directions and directions.*A* - The
**vector sum**+*A*it has as a module the sum of the modules of both, the same direction and the same direction of the given vectors.*B*

### Sum of two vectors with the same direction and the opposite direction

Sean ** A** y

**:**

*B*To obtain the sum vector we follow the following steps:

- We draw the vector
below the vector*B*, so that it is consecutive, respecting its modules, directions and directions.*A* - The vector adds
+*A*it has as a module the difference of the modules of both, the same direction and the direction of the greater vector.*B*

### Sum of two vectors with different directions

To add two vectors ** A** y

**forming an angle to each other, two methods are used: the triangle method and the parallelogram method.**

*B***Triangle method**

- We draw the vectors consecutively, that is, the origin of
has to match the end*B*.*A* - The vector adds
+*A*has as its origin the origin of*B*and as an extreme, that of*A*.*B*

### Parallelogram law or method

- We draw the vector
at the origin of a Cartesian plane respecting its module, direction and direction.*A* - We draw at the origin of
, the vector*A***B**respecting its module, direction and sense. - Lines are drawn parallel to each vector forming a parallelogram.
- The resulting vector will be the diagonal of the parallelogram starting at the origin of the Cartesian plane.

**Properties of the sum vector**

**Commutative property**

Let there be two vectors ** A** y

**; according to the parallelogram law, the diagonal AC represents the vector sum**

*B***:**

*S*According to the figure:

** S **=

**+**

*A***=**

*B***+**

*B*

*A*The addition of vectors is commutative.

**Associative property**

Let there be three vectors ** A**,

**y**

*B***and the vector adds**

*C***=**

*S***+**

*A***+**

*B*

*C*

According to the figure:

** AB **=

**+**

*AD***→**

*DB***=**

*S***+ (**

*A***+**

*B***)**

*C*

Y

** AB **=

**+**

*AC***→**

*CB***= (**

*S***+**

*A***) +**

*B*

*C*Thus

** A **+ (

**+**

*B***) = (**

*C***+**

*A***) +**

*B*

*C*

The sum of vectors is associative.

**Distributive property**

The sum of vectors is distributive with respect to the multiplication by scales:

(** A **+

**) μ = μ**

*B***+ μ**

*A*

*B*