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Vectors and their Operations: Vector operations using Cartesian vector notation

Planar vector operations using CVN (two dimensions)

Addition of several vectors \bold F_1 + \bold F_2 + \dots +\bold F_n using CVN takes the following steps:

1- Express each vector in CVN by resolving the vector to its scalar components:

    \[\begin{split} \bold F_1&=F_{1x}\bold i + F_{1y}\bold j\\ \bold F_2&=F_{2x}\bold i + F_{2y}\bold j\\ &\vdots\\ \bold F_n&=F_{nx}\bold i + F_{ny}\bold j \end{split} \]

2- Add the respective scalar components (components on the same axis):

    \[\begin{split} \bold R_x&= (F_{1x}+F_{2x} + \dots + F_{nx})\bold i =\left(\sum_i^nF_{ix}\right)\bold i= R_x\bold i\\ \bold R_y&= (F_{1y}+F_{2y} + \dots + F_{ny})\bold j = \left(\sum_i^nF_{iy}\right)\bold j=R_y\bold j \end{split} \]

in which \bold R_x and \bold R_y are the Cartesian vector components of the resultant vector \bold R.

The above steps can be summarized as:

(2.7)   \[\bold R=(\sum F_x)\bold i + (\sum F_y)\bold j  \]

in which \sum F_x and \sum F_y represent the algebraic sums of the scalar components along the x and y axes respectively.

3- Form the resultant vector \bold R = \bold R_x +\bold R_y. The magnitude of \bold R and its direction with respect to the x axis can be obtained by,

    \[\begin{split}|\bold R|^2&=R_x^2 + R_y^2 \quad \text{or}\quad |\bold R| =\sqrt {R_x^2 + R_y^2}\\\tan \theta &= \frac{R_y}{R_x},\quad 0^\circ\le \theta < 360^\circ\\&\text{or}\\\tan \alpha &= \frac{| R_y|}{| R_x|}, \quad 0^\circ\le \alpha < 90^\circ\end{split}\]

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 z direction. This means,

(2.8)   \[ \bold R=R_x\bold i + R_y\bold j + R_z\bold k = (\sum F_x)\bold i + (\sum F_y)\bold j + (\sum F_z)\bold kj  \]

in which \sum F_x, \sum F_y, and \sum F_z represent the algebraic sums of the scalar components along the x, y and z axes respectively.

The magnitude of the resultant vector is,

(2.9)   \[ |\bold R|=\sqrt{R_x^2 +R_y^2 +R_z^2} \]

The direction of \bold R can be expressed by the coordinate direction angles. The angles are determined using Eq 2.5 as,

    \[ \cos \alpha=\frac{R_x}{|\bold R|} \quad \cos \beta=\frac{R_y}{|\bold R|} \quad \cos \gamma=\frac{R_z}{|\bold R|}\]