# Vector sum 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 B, we must draw them consecutively (respecting their modules, directions and directions), making the origin of B  match end A; the vector adds A + B will have as its origin, the origin A and as an extreme the origin of B.

## Definition

Let there be two vectors A y B, he vector sum obtained by adding the vector A with vector BThat is, by adding the components of each vector:

A + B = (Ax + Bx, TOy + By, TOz + Bz)

Example: Sean A = (3, 2, -4) and B = (-3, 2, 7), calculate the vector A + B.

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

### 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:

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

1. We draw the vector B below the vector A, so that it is consecutive, respecting its modules, directions and directions.
2. The vector adds A + B it has as a module the difference of the modules of both, the same direction and the direction of the greater vector. ### Sum of two vectors with different directions

To add two vectors A  y B forming an angle to each other, two methods are used: the triangle method and the parallelogram method.

### Triangle method

1. We draw the vectors consecutively, that is, the origin of B has to match the end A.
2. The vector adds A + B has as its origin the origin of A  and as an extreme, that of B. ### Parallelogram law or method

1. We draw the vector A at the origin of a Cartesian plane respecting its module, direction and direction.
2. We draw at the origin of A, the vector B respecting its module, direction and sense.
3. Lines are drawn parallel to each vector forming a parallelogram.
4. 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 B; according to the parallelogram law, the diagonal AC represents the vector sum S:

According to the figure:

S = A + B = B + A

The addition of vectors is commutative.

### Associative property Let there be three vectors A, B y C and the vector adds 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 + B) μ = μA + μB