Question

(a) In unit-vector notation, what is the sum $\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}$ if $\overrightarrow{\mathbf{a}}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}\mathbf{)}\hat{\mathbf{i}}\mathbf{}\mathbf{+}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}\mathbf{)}\hat{\mathbf{j}}$and $\overrightarrow{\mathbf{b}}\mathbf{=}\mathbf{(}\mathbf{}\mathbf{-}\mathbf{13}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}\mathbf{)}\mathbf{}\hat{\mathbf{i}}\mathbf{+}\mathbf{(}\mathbf{7}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}\mathbf{)}\mathbf{}\hat{\mathbf{j}}$? What are the (b) magnitude and (c) direction of $\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}$?

Short answer

(a) Sum is $(-9.0\mathrm{m})\hat{\mathrm{i}}+(10\mathrm{m})\hat{\mathrm{j}}.$

(b) Magnitude is $13\mathrm{m}$.

(c) Direction is ${132}^{°}$.

Step-by-step solution

Given information

The given two vectors are,

$$\begin{gathered} \overrightarrow{\mathrm{a}}=(4.0\mathrm{m})\hat{\mathrm{i}}+(3.0\mathrm{m})\hat{\mathrm{j}} \\ \overrightarrow{\mathrm{b}}=(-13.0\mathrm{m})\hat{\mathrm{i}}+(7.0\mathrm{m})\hat{\mathrm{j}} \end{gathered}$$

To understand the concept of vector addition

A vector quantity has magnitude as well as direction. To add two vector quantities, we should use the vector law of addition. If two vectors are exactly in the same direction, we can add them using normal addition rules. But if the directions of two vectors are different, then we need to find the component of the vectors along the unit vector direction and then add them.

In the problem, vector quantities are given in terms of unit vectors. Therefore, we can add them using the vector law of addition. Using the resultant vector in terms of the unit vector, we can find the direction of the resultant vector.

Formula:

Considering the sum of $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$is$\overrightarrow{\mathrm{r}}$, the magnitude of the vector is given by,

localid="1657014645424" $\mathrm{r}=\sqrt{{{\mathrm{r}}_{\mathrm{x}}}^2+{{\mathrm{r}}_{\mathrm{y}}}^2}.........\left(\mathrm{i}\right)$

The direction is given by,

localid="1657014745273" $\mathrm{\theta }={\tan }^{-1}\left(\frac{{\mathrm{r}}_{\mathrm{y}}}{{\mathrm{r}}_{\mathrm{x}}}\right)........\left(\mathrm{ii}\right)$

(a) To find the sum of a→+b→ 

The x, y components of $\overrightarrow{r}$are

role="math" localid="1657015061990" $$\begin{gathered} {\mathrm{r}}_{\mathrm{x}}={\mathrm{a}}_{\mathrm{x}}+{\mathrm{b}}_{\mathrm{x}} \\ =\left(4\mathrm{m}\right)\hat{\mathrm{i}}+\left(-13\mathrm{m}\right)\hat{\mathrm{i}} \\ =\left(-9\mathrm{m}\right)\hat{\mathrm{i}} \end{gathered}$$

And

$$\begin{gathered} {\mathrm{r}}_{\mathrm{y}}={\mathrm{a}}_{\mathrm{y}}+{\mathrm{b}}_{\mathrm{y}} \\ =\left(3\mathrm{m}\right)\hat{\mathrm{j}}+\left(7\mathrm{m}\right)\hat{\mathrm{j}} \\ =\left(10\mathrm{m}\right)\hat{\mathrm{j}} \end{gathered}$$

Thus,

$$\begin{gathered} \overrightarrow{\mathrm{r}}={\mathrm{r}}_{\mathrm{x}}+{\mathrm{r}}_{\mathrm{y}} \\ =\left(-9\mathrm{m}\right)\hat{\mathrm{i}}+\left(10\mathrm{m}\right)\hat{\mathrm{j}} \end{gathered}$$

(b) To find magnitude of a→+b→ 

Using equation (i), the magnitude will be,

$\mathrm{r}=\sqrt{{{\mathrm{r}}_{\mathrm{x}}}^2+{{\mathrm{r}}_{\mathrm{y}}}^2}$

Substitute the values in the above equation,

$$\begin{gathered} \left|\overrightarrow{\mathrm{r}}\right|=\sqrt{\left(-9\mathrm{m}\right)+{\left(10\mathrm{m}\right)}^2} \\ =13\mathrm{m} \end{gathered}$$

Thus, the magnitude is $13\mathrm{m}$

(c) To find magnitude of a→+b→ 

Using equation (ii), the magnitude is given by,

$\mathrm{\theta }={\tan }^{-1}\left(\frac{10\hspace{0.33em}\mathrm{m}}{-9\mathrm{m}}\right)$

Thus, the magnitude is $-{48}^{°}$or${132}^{°}$ .

Since the x component of the resultant is negative and y component is positive, it is found the magnitude of the given vector to be${132}^{°}$ .