Addition and Subtraction of Vectors


 Addition and Subtraction of Vectors

\(\blacksquare\) A resultant vector is the combination of two or more single vectors.
Parallel vectors

(a) The addition of two parallel vectors \(\utilde{a}\) and \(\utilde{b}\) can be combined to produce a resultant vector represented as \(\utilde{a}+\utilde{b}\).

For example,



(b) The subtraction of vector \(\utilde{b}\) from vector \(\utilde{a}\) is the sum of vector \(\utilde{b}\) and negative vector \(\utilde{b}\), that is

For example,


\(\blacksquare\) The resultant vector can be obtained using
triangle law

parallelogram law

polygon law

Example 1:

It is given that

\(\begin{aligned} \overrightarrow{CD}&=3\utilde{r}, \\\\ \overrightarrow{EF}&=5\utilde{r} ,\\\\ \overrightarrow{FG}&=\utilde{r}. \end{aligned}\)


If \(|\utilde{r}|=3 \text{ units}\), find the magnitude of the following expression:

\(\begin{aligned} 2\overrightarrow{CD}+\overrightarrow{EF}-3\overrightarrow{FG} \end{aligned}\)

Express the expression in term of \(\utilde{r}\).


\(\begin{aligned} &2\overrightarrow{CD}+\overrightarrow{EF}-3\overrightarrow{FG} \\\\ &=2(3\utilde{r})+5 \utilde{r}-3(\utilde{r}) \\\\ &=6\utilde{r}+5 \utilde{r}-3\utilde{r}. \end{aligned}\)


Since \(|\utilde{r}|=3 \text{ units}\),


\(\begin{aligned} &|6\utilde{r}+5 \utilde{r}-3\utilde{r}|\\\\ &=|8\utilde{r}|\\\\&=8|\utilde{r}|\\\\ &=8(3) \\\\ &=24 \text{ units}. \end{aligned}\)

Example 2:

Two forces \(\bold{F}_1=50 \text{ N}\) and \(\bold{F}_2=30 \text{ N}\) are acting on a moving object.

Determine the magnitude of the force and the direction of the movement of the object if

\(\bold{F}_1\) and \(\bold{F}_2\) are in the opposite direction.


If \(\bold{F}_1\) and \(\bold{F}_2\) are in the opposite direction,





\(\begin{aligned} \bold{F}_1+(-\bold{F}_2)&=50+(-30)\\\\&=50-30 \\\\ &=20\text{ N}. \end{aligned}\)


The magnitude of the force acted on the object is \(20\text{ N}\) in the same dircetion as \(\bold{F}_1\).