Vector sum

vector sumThe 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:

add vectorsTo 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.

add vectors

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

Sean A y B:

add vectors

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.

add vectors with opposite sense

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.

Add vectors (triangle method)

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.

add vectors (parallelogram method)

Properties of the sum vector

Commutative property

add vectorsLet 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

add vectorsLet 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