## Addition and Subtraction of Vectors

 8.2 Addition and Subtraction of Vectors

Addition and Subtraction of Parallel Vector

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:

 Subtraction of Two Parallel Vector

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

For example:

Addition and Subtraction of Non-Parallel Vector
 Triangle Law

 Parallelogram Law

 Polygon Law

Example $$1$$
 Question

It is given that,

$$\overrightarrow{CD}=3\utilde{r}$$,
$$\overrightarrow{EF}=5\utilde{r}$$,
$$\overrightarrow{FG}=\utilde{r}$$.

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

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

 Solution

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 $$|\utilde{r}|=3$$ units, hence:

\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 $$2$$
 Question

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.

 Solution

If $$\bold{F}_1$$ and $$\bold{F}_2$$ are in the opposite direction, then:

Magnitude:

\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 $$\bold{F}_1$$.

## Addition and Subtraction of Vectors

 8.2 Addition and Subtraction of Vectors

Addition and Subtraction of Parallel Vector

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:

 Subtraction of Two Parallel Vector

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

For example:

Addition and Subtraction of Non-Parallel Vector
 Triangle Law

 Parallelogram Law

 Polygon Law

Example $$1$$
 Question

It is given that,

$$\overrightarrow{CD}=3\utilde{r}$$,
$$\overrightarrow{EF}=5\utilde{r}$$,
$$\overrightarrow{FG}=\utilde{r}$$.

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

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

 Solution

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 $$|\utilde{r}|=3$$ units, hence:

\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 $$2$$
 Question

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.

 Solution

If $$\bold{F}_1$$ and $$\bold{F}_2$$ are in the opposite direction, then:

Magnitude:

\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 $$\bold{F}_1$$.