91Ó°ÊÓ

Here, vector law of addition and subtraction is used to find the resultant of the given vector. Further using the general formula for the magnitude and the angle, the magnitude and the angle of the given vector can be calculated.

Formulae

$$\begin{gathered} \stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}=\left(a_x\right)\stackrel{Áåœ}{\mathrm{i}}+\left(a_y\right)\stackrel{Áåœ}{j}+\left(b_x\right)\stackrel{Áåœ}{\mathrm{i}}+\left(b_y\right)\stackrel{Áåœ}{j}+\left(c_x\right)\stackrel{Áåœ}{\mathrm{i}}+\left(c_y\right)\stackrel{Áåœ}{j} \\ \stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}=\left(a_x+b_x+c_x\right)\stackrel{Áåœ}{\mathrm{i}}+\left(a_y+b_y+c_y\right)\stackrel{Áåœ}{j} \\ \stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}=\left(a_x+b_x\right)\stackrel{Áåœ}{\mathrm{i}}+\left(a_y+b_y\right)\stackrel{Áåœ}{\mathrm{j}} \\ r=\left|\stackrel{\mathit{→}}{\mathit{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}\right|=\sqrt{{\left(a_x+b_x+c_x\right)}^2+{\left(a_y+b_y+c_y\right)}^2} \\ \mathrm{θ}={\tan }^{-1}\left(\frac{a_x+b_x+c_x}{a_y+b_y+c_y}\right) \\ \mathrm{Given}\mathrm{are} \\ \stackrel{\mathit{→}}{\mathit{a}}=\left(50\mathrm{m}\right)\cos \left(30\right)\stackrel{Áåœ}{\mathrm{i}}+\left(50\mathrm{m}\right)\sin \left(30\right)\stackrel{Áåœ}{\mathrm{j}} \\ \stackrel{→}{\mathrm{b}}=\left(50\mathrm{m}\right)\cos \left(195\right)\stackrel{Áåœ}{\mathrm{i}}+\left(50\mathrm{m}\right)\sin \left(195\right)\stackrel{Áåœ}{\mathrm{j}} \\ \stackrel{→}{\mathrm{c}}=\left(50\mathrm{m}\right)\cos \left(315\right)\stackrel{Áåœ}{\mathrm{i}}+\left(50\mathrm{m}\right)\sin \left(315\right)\stackrel{Áåœ}{\mathrm{j}} \end{gathered}$$

Using the above values the vector $\stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}$can be written as

$$\begin{gathered} \stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}=\left(30.4\right)\stackrel{Áåœ}{i}\left(-23.3m\right)\stackrel{Áåœ}{j} \\ \mathrm{Magnitude}\mathrm{of}\stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}\mathrm{is} \\ \left|\stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}\right|=\sqrt{\left(30.4{\mathrm{m}}^2\right)+{\left(-23.3\mathrm{m}\right)}^2} \\ \left|\stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}\right|=38\mathrm{m} \end{gathered}$$

The angle between $\stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}$and x axis is

${\tan }^{-1}\left(\frac{-23.3\mathrm{m}}{30.4\mathrm{m}}\right)=-37.5°$

$$\begin{gathered} \stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}=\left(127\mathrm{m}\right)\stackrel{Áåœ}{\mathrm{i}}+\left(2.60\mathrm{m}\right)\stackrel{Áåœ}{\mathrm{j}} \\ \left|\stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}\right|=\sqrt{{\left(127\mathrm{m}\right)}^2+{\left(2.60\mathrm{m}\right)}^2} \\ \left|\stackrel{→}{\mathrm{a}}+\stackrel{→}{\mathrm{b}}+\stackrel{→}{\mathrm{c}}\right|=1.30×{10}^2\mathrm{m} \end{gathered}$$

The angle between$\stackrel{\mathit{→}}{\mathit{a}}\mathit{+}\stackrel{\mathit{→}}{\mathit{b}}\mathit{+}\stackrel{\mathit{→}}{\mathit{c}}$and x axis is

${\tan }^{-1}\left(\frac{26.5\mathrm{m}}{127\mathrm{m}}\right)=1.2°$

Therefore, the angle between $\stackrel{\mathit{→}}{a}\mathit{+}\stackrel{\mathit{→}}{b}\mathit{+}\stackrel{\mathit{→}}{c}$ and +x axis is $1.2°$

$$\begin{gathered} \stackrel{→}{d}=\stackrel{\mathit{→}}{a}\mathit{+}\stackrel{\mathit{→}}{b}\mathit{+}\stackrel{\mathit{→}}{c}\mathit{=}\left(-40.4\mathrm{m}\right)\stackrel{Áåœ}{\mathrm{i}}+\left(47.4\mathrm{m}\right)\stackrel{Áåœ}{\mathrm{j}} \\ \left|\stackrel{→}{\mathrm{d}}\right|=\sqrt{{\left(-40.4\mathrm{m}\right)}^2+{\left(47.4\mathrm{m}\right)}^2}=62\mathrm{m} \\ \left|\stackrel{→}{\mathrm{d}}\right|=62\mathrm{m} \end{gathered}$$

The angle between$\stackrel{→}{d}$and +x axis is

${\tan }^{-1}\left(\frac{47.4\mathrm{m}}{-40.4\mathrm{m}}\right)=50°$

As vector$\stackrel{→}{d}$is in third quadrant so,

$180°-50°=130°$

Therefore, the angle between$\stackrel{→}{d}$and +x axis is