Question

The following forces act on a particle P:F1=2i+3j-5k, F2=-5i+j+3k, F3=i-2j+4k, F4=4i-3j-2k, measured in pounds. Find:
a) the resultant (sum) of the forces
b) the magnitude of the resultant

Answer 1

$\textbf{Given:}$

$\text{The forces acting on the particle P are}$

$\overrightarrow{F_1}=2\overrightarrow{i}+3\overrightarrow{j}-5\overrightarrow{k}$

$\overrightarrow{F_2}=-5\overrightarrow{i}+\overrightarrow{j}+3\overrightarrow{k}$

$\overrightarrow{F_3}=\overrightarrow{i}-2\overrightarrow{j}+4\overrightarrow{k}$

$\overrightarrow{F_4}=4\overrightarrow{i}-3\overrightarrow{j}-2\overrightarrow{k}$

$\textbf{Resultant force }\bf\,\overrightarrow{F}=\overrightarrow{F_1}+\overrightarrow{F_2}+\overrightarrow{F_3}+\overrightarrow{F_4}$

$\implies\,\overrightarrow{F}=2\overrightarrow{i}-\overrightarrow{j}+0\overrightarrow{k}$

$\textbf{Magnitude of the resultant force }\bf\overrightarrow{F}$

$=|\overrightarrow{F}|$

$=\sqrt{2^2+(-1)^2+0^2}$

$=\sqrt{4+1+0}$

$=\sqrt{5}$

$\therefore\textbf{The resultant force is }\bf\,2\overrightarrow{i}-\overrightarrow{j}+0\overrightarrow{k}\:\textbf{ and its magnitude is }\bf\,\sqrt{5}$

Answer 2

Answer:

\textbf{Given:}Given:

\text{The forces acting on the particle P are}The forces acting on the particle P are

\overrightarrow{F_1}=2\overrightarrow{i}+3\overrightarrow{j}-5\overrightarrow{k}F1=2i+3j−5k

\overrightarrow{F_2}=-5\overrightarrow{i}+\overrightarrow{j}+3\overrightarrow{k}F2=−5i+j+3k

\overrightarrow{F_3}=\overrightarrow{i}-2\overrightarrow{j}+4\overrightarrow{k}F3=i−2j+4k

\overrightarrow{F_4}=4\overrightarrow{i}-3\overrightarrow{j}-2\overrightarrow{k}F4=4i−3j−2k

\textbf{Resultant force }\bf\,\overrightarrow{F}=\overrightarrow{F_1}+\overrightarrow{F_2}+\overrightarrow{F_3}+\overrightarrow{F_4}Resultant force F=F1+F2+F3+F4

\implies\,\overrightarrow{F}=2\overrightarrow{i}-\overrightarrow{j}+0\overrightarrow{k}⟹F=2i−j+0k

\textbf{Magnitude of the resultant force }\bf\overrightarrow{F}Magnitude of the resultant force F

=|\overrightarrow{F}|=∣F∣

=\sqrt{2^2+(-1)^2+0^2}=22+(−1)2+02

=\sqrt{4+1+0}=4+1+0

=\sqrt{5}=5

\therefore\textbf{The resultant force is }\bf\,2\overrightarrow{i}-\overrightarrow{j}+0\overrightarrow{k}\:\textbf{ and its magnitude is }\bf\,\sqrt{5}∴The resultant force is 2i−j+0k and its magnitude is 5

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