# ## Vectors and their Operations: Vector operations using Cartesian vector notation

### Planar vector operations using CVN (two dimensions)

Addition of several vectors using CVN takes the following steps:

1- Express each vector in CVN by resolving the vector to its scalar components: 2- Add the respective scalar components (components on the same axis): in which and are the Cartesian vector components of the resultant vector .

The above steps can be summarized as:

(2.7) in which and represent the algebraic sums of the scalar components along the and axes respectively.

3- Form the resultant vector . The magnitude of and its direction with respect to the axis can be obtained by, Remark: the apparent location of a vector on a plane does not affect its CVN.

Planar vector addition using CVN is illustrated by the following interactive tool.

### Spatial vector Addition using CVN (three dimensions)

Once the vectors to be summed are resolved into their components and represented in CVN, the similar steps as in the coplanar case should be followed but with including components in the direction. This means,

(2.8) in which , , and represent the algebraic sums of the scalar components along the , and axes respectively.

The magnitude of the resultant vector is,

(2.9) The direction of can be expressed by the coordinate direction angles. The angles are determined using Eq 2.5 as, 