Vector Components Diagram

In the previous section, vectors were described using its magnitude and direction. This makes mathematical operations difficult (cannot simply add angles). To help solve this problem, vectors are usually split into components. For example, if two vectors, F₁ and F₂, give the vector F when added together, then F₁ and F₂ are said to be components of the vector F:

     F = F₁ + F₂

If the vectors F₁ and F₂ are perpendicular to each other, they are called rectangular, or Cartesian, components. If the x-y axis is oriented along these components, they are labeled F_x and F_y respectively:

     F = F_x + F_y

If two vectors, i and j, have a magnitude of one and are in the x and y direction respectively, then F can be written as

     F = F_xi + F_yj

The vectors i and j are called Cartesian unit vectors, and F_x and F_y are the scalar components of the vector F. This configuration is the most common to describe a vector and allows vectors to be added and subtracted quickly and easily.

If the angle θ is measured from the x axis counterclockwise, then the scalar components are

     F_x = F cosθ
     F_y = F sinθ

Unit vectors in the x and y directions (i and j) where used the above paragraphs, but unit vectors can also be used in any direction. Unit vectors give direction where magnitude gives the length of the vector. Unit vectors are defined as

     u_F = F/F

and have a magnitude of 1.

Vector Angle Relationship

When a set of vectors are described using Cartesian unit vectors (i and j), then the resultant vector is just the addition (or subtraction) each components, giving

     F_R = F₁ + F₂ +... = ΣF_xi + ΣF_yj

       where ΣF_x = F_1x + F_2x +...
                 ΣF_y = F_1y + F_2y +...

Note, F_n-x and F_n-y are the components of each individual vector where n is the numbering of each vector being added.

The magnitude and direction of the resultant vector, F_R, can be determined just like any individual vector by,