Here are two vectors:

$\overrightarrow{\mathbf{a}}\mathbf{=}\left(4.00m\right)\stackrel{\mathbf{⏜}}{\mathbf{i}}\mathbf{-}\left(3.00m\right)\stackrel{\mathbf{⏜}}{\mathbf{j}}\overrightarrow{\mathbf{b}}\mathbf{=}\left(6.00m\right)\mathit{i}\mathbf{+}\left(8.00m\right)\mathit{j}$

What are (a) the magnitude and (b) the angle (relative to i ) of $\overrightarrow{\mathbf{a}}$? What are (c) the magnitude and (d) the angle of $\overrightarrow{\mathbf{b}}$? What are (e) the magnitude and (f) the angle of $\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}$;(g) the magnitude and (h) the angle of $\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{a}}$; and (i) the magnitude and (j) the angle ofrole="math" localid="1656943686601" $\overrightarrow{\mathbf{a}}\mathbf{-}\overrightarrow{\mathbf{b}}$? (k) What is the angle between the directions of $\overrightarrow{\mathbf{b}}\mathbf{-}\overrightarrow{\mathbf{a}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\overrightarrow{\mathbf{a}}\mathbf{-}\overrightarrow{\mathbf{b}}$?

$$\begin{gathered} a⃗=(4.00m)\stackrel{⏜}{i}-(3.00m)\stackrel{⏜}{j} \\ \overrightarrow{b}=(6.00\mathrm{m})\mathrm{i}+(8.00\mathrm{m})j \end{gathered}$$

Find the magnitude of any vector by using its values and perform different vector operations using vector algebra. Also,find the angle between vectors and axis usingtrigonometry.

Formulae are as follow:

$\mathrm{Magnitude}\mathrm{of}\overrightarrow{\mathrm{a}}\mathrm{is}\mathrm{a}\sqrt{{\left(a_x\right)}^2+{\left(a_y\right)}^2}$

$$\begin{gathered} \mathrm{Angle}\mathrm{between}\overrightarrow{a}\mathrm{and}\mathrm{X}\mathrm{axis}\mathrm{is}\mathrm{tan\theta }=\left(\frac{{\mathrm{a}}_{\mathrm{y}}}{{\mathrm{a}}_{\mathrm{x}}}\right) \\ \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=\left({\mathrm{a}}_{\mathrm{x}}+{\mathrm{b}}_{\mathrm{x}}\right)\stackrel{⏜}{\mathrm{i}}+\left({\mathrm{a}}_{\mathrm{y}}+{\mathrm{b}}_{\mathrm{y}}\right)\stackrel{⏜}{\mathrm{j}} \\ \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}=\left({\mathrm{a}}_{\mathrm{x}}-{\mathrm{b}}_{\mathrm{x}}\right)\stackrel{⏜}{\mathrm{i}}+\left({\mathrm{a}}_{\mathrm{y}}+{\mathrm{b}}_{\mathrm{y}}\right)\stackrel{⏜}{\mathrm{j}} \end{gathered}$$

where,a,b are vectors and $\theta$is the angle between$a_{\mathrm{x}}\mathrm{and}{a}_y$.

The magnitude of$\overrightarrow{a}$ is,

$$\begin{gathered} a=\sqrt{{\left(4.0\right)}^2+{\left(-3.0\right)}^2} \\ =5.0\mathrm{m} \end{gathered}$$

Hence, magnitude of is 5.0 m .

The angle between$\overrightarrow{a}$and X axis is,

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ ={\tan }^{-1}\left(\frac{-3.0}{4.0}\right) \\ =-37° \end{gathered}$$

Hence, the vector is 37⁰ clockwise from X axis.

The magnitude of $\overrightarrow{b}$is,

$$\begin{gathered} b=\sqrt{{\left(6.0\right)}^2+{\left(8.0\right)}^2} \\ =10.0\mathrm{m} \end{gathered}$$

Hence,magnitude$\overrightarrow{b}$ of is 10.0m .

The angle between $\overrightarrow{b}$and X axis is,

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ ={\tan }^{-1}\left(\frac{8.0}{6.0}\right) \\ =53° \end{gathered}$$

Hence, angle of $\overrightarrow{b}$is $53°$.

Use vector addition law to find the sum$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$

$$\begin{gathered} \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=\left(6.0+4.0\right)\stackrel{⏜}{i}+\left(\left(8.0\right)+\left(-3.0\right)\right)\stackrel{⏜}{j} \\ =\left(10\right)\stackrel{⏜}{i}+\left(5\right)\stackrel{⏜}{j} \end{gathered}$$

Magnitude of$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ is calculated as,

$$\begin{gathered} \left|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\right|=\sqrt{{10}^2+5.0^2} \\ =11.0 \end{gathered}$$

Hence,magnitude ofrole="math" localid="1656946159137" $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ is 11m .

Angle of $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$with X axis is,

role="math" localid="1656946653705" $$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{5.0}{10.0}\right) \\ =27° \end{gathered}$$

Hence, angle of $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$is$27°$.

Use vector subtraction law to find the $\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$

$$\begin{gathered} \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}=\left(6.0-4.0\right)\stackrel{⏜}{i}+\left(8.0+3.0\right)\stackrel{⏜}{j} \\ =\left(2.0\right)\stackrel{⏜}{i}+\left(11.0\right)\stackrel{⏜}{j} \end{gathered}$$

The magnitude can be found as,

role="math" localid="1656946628062" $$\begin{gathered} \left|\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}\right|=\sqrt{2^2+{11}^2} \\ =11 \end{gathered}$$

Hence, magnitude of $\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$ is 11m .

The angle of$\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$ with X axis is,

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{11.0}{2.0}\right) \\ =80° \end{gathered}$$

Hence,angle of $\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$is$80°$ .

$$\begin{gathered} \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}=\left(4.0-6.0\right)\stackrel{⏜}{i}+\left(\left(-3.0\right)-\left(8.0\right)\right)\stackrel{⏜}{j} \\ =\left(-2.0\right)\stackrel{⏜}{i}+\left(-11\right)\stackrel{⏜}{j} \\ \mathrm{Magnituted}\mathrm{of}\overrightarrow{a}\mathit{}\mathit{-}\overrightarrow{b}\mathrm{is}, \\ \left|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}\right|=\sqrt{-2^2+\left(-11\right)\stackrel{⏜}{\mathrm{j}}} \\ =11.0 \\ \mathrm{Hence},\mathrm{magnitude}\mathrm{of}\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}\mathrm{is}11\mathrm{m}. \end{gathered}$$

Angle of $\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$with X axis is,

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{11.0}{2.0}\right) \\ =80° \end{gathered}$$

Hence, angle of is $80°$.

Since, this vector lies in Quadrant III, its angle with positive X axis is 80⁰+180⁰ = 260⁰.

Since,$\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}=-1\left(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}\right)$, they are directed opposite to each other.

Hence, the angle between$\overrightarrow{b}-\overrightarrow{a}$and $\overrightarrow{\mathrm{a}}-\overrightarrow{b}$is $180°$

Therefore,by using laws of vector algebra, add and subtract the vectors. It is also possible to find the magnitude of vectors and their direction with particular axis.