Question

Three vectors $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\overrightarrow{\mathbf{c}}$each have a magnitude of 50mand lie in an xy plane. Their directions relative to the positive direction of the x axis are $\mathbf{30}\mathbf{°}$,$\mathbf{195}\mathbf{°}$, and $\mathbf{315}\mathbf{°}$, respectively. What are (a) the magnitude and (b) the angle of the vector localid="1656259759790" $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}$, and (c) the magnitude and (d) the angle of $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}}$ ? What are the (e) magnitude and (f) angle of a fourth vector $\overrightarrow{\mathbf{d}}$ such that $\left(\overrightarrow{a}+\overrightarrow{b}\right)\mathit{-}\mathit{}\left(\overrightarrow{c}+\overrightarrow{d}\right)\mathit{=}\mathit{0}$?

Short answer

Magnitude of vector $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$is

Angle of vector $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$is

Magnitude of vector$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$ is

Angle of vector$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}$ is

Magnitude of fourth vector is

Angle of fourth vector is

Step-by-step solution

To understand the concept

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

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

To find magnitude of vector a→+b→+c→

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

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

To find the angle between vector a→+b→+c→ and x axis

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

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

This is equivalent to $37.5°$clockwise from the +x axis and$322.5°$counterclockwise from +x axis

To find magnitude of vector a→+b→+c→

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

To find the angle between vector a→+b→+c→ and x axis

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

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

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

To find magnitude of fourth vector d→

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

To find the angle between vector d→ and x axis

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

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

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

$180°-50°=130°$

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