Question

(a) What is the sum of the following four vectors in unit -vector notation? For the sum. What is b) the magnitude, c) the angle in degrees, and d) the angle in radians? Positive angles are counter clockwise from the positive direction of the x- axis; negative angles are clockwise.
$\overrightarrow{E}:6.00m$ at $+0.900rad$
$\overrightarrow{F}:5.00m$ at $-75.0^o$
$\overrightarrow{G}:4.00m$ at $1.20rad$
$\overrightarrow{H:}6.00m$ at $-210^o$

Answer 1

Hint: In a scalar unit if the value of physical functions at each point of a field is a scalar quantity then it is a scalar field like temperature of atmosphere, depth of sea water from surfaces etc. In a vector field if the value of function at each point of a field is vector quantity then it is called a vector field like wind velocity of atmosphere , the force of gravity on a mass in space ,forces on a charged body placed in an electric field etc.

Complete step-by-step answer:
Field is a region where each point has a corresponding value of some physical function. For example, the electric field in a region has some specific direction of the $\overrightarrow{E}$ component at different points. Electric field (E) is a vector quantity.
On the sphere the points P(X, Y, Z)
${{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{R}^{2}}$
A unit vector which is perpendicular to the sphere radially outwards is given by:
$\begin{align}
  & \widehat{N}=\dfrac{x}{\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}}\widehat{i}+\dfrac{y}{\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}}\widehat{j}+\dfrac{Z}{\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}}\widehat{k} \\
 & \widehat{N}=\dfrac{x}{R}\widehat{i}+\dfrac{y}{R}\widehat{j}+\dfrac{z}{R}\widehat{k} \\
\end{align}$
a) The sum of the following four vectors is:
\[{{\overrightarrow{V}}_{sum}}=\overrightarrow{E}+\overrightarrow{F}+\overrightarrow{G}+\overrightarrow{H}\]
After substituting their respective values
\[\begin{align}
  & {{\overrightarrow{V}}_{sum}}=(6+4+5+6)mat+(0.9+1.2)rad+{{(-75-120)}^{0}} \\
 & {{\overrightarrow{V}}_{sum}}=21mat+2.1rad-{{285}^{0}} \\
\end{align}\]
b) The magnitude of the four vectors is:
$\begin{align}
  & \left| {{\overrightarrow{V}}_{R}} \right|=\sqrt{{{(2.1)}^{2}}+{{(2.1)}^{2}}+{{(-285)}^{2}}} \\
 & \left| {{\overrightarrow{V}}_{R}} \right|=\sqrt{{{(285.780)}^{2}}} \\
 & \left| {{\overrightarrow{V}}_{R}} \right|=285.78 \\
\end{align}$
c) The angle in degrees is :
$\begin{align}
  & \theta ={{\cos }^{-1}}\left[ \dfrac{-285}{285.78} \right] \\
 & \theta =175.76 \\
\end{align}$
d) The angle in radians is:
$\phi ={{\cos }^{-1}}\left[ \dfrac{2.1}{285.78} \right]=89.57$

Note: Students unit vector is the ratio of vector itself by its magnitude and Gauss’s diversion theorem which states that volume integral of the divergence of vector field $\overrightarrow{A}$.taken over any volume V is equal to surface integral of $\overrightarrow{A}$ taken over the closed surface that bonds the volume V $\int_{V}{(\nabla }\cdot \overrightarrow{A}).dV=\oint\limits_{S}{\overrightarrow{A}\cdot \overrightarrow{dS}}$.