Question

Three force $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{R}$, each of $15$ units, act along $AB,BC$ and $CA$ respectively. The position vectors of $A,B$ and $C$ are $\overrightarrow{OA}=\overrightarrow{2i}-\overrightarrow{j}+\overrightarrow{3k}$, $\overrightarrow{OB}=\overrightarrow{5i}+\overrightarrow{3j}-\overrightarrow{2k}$ and $\overrightarrow{OC}=-\overrightarrow{2i}+\overrightarrow{2j}+\overrightarrow{3k}$ respectively. The resultant force vector is

• A. $(12+\frac{9}{\sqrt{3}})\overrightarrow{i}-(9-6\sqrt{2}+\sqrt{3})\overrightarrow{j}+(5\sqrt{3}-\frac{15}{\sqrt{2}})\overrightarrow{k}$
• B. $(12+\frac{9}{\sqrt{3}})\overrightarrow{i}+(9-6\sqrt{2}+\sqrt{3})\overrightarrow{j}+(\frac{15}{\sqrt{2}}-5\sqrt{3})\overrightarrow{k}$
• C. $\overrightarrow{75i}+\overrightarrow{60j}+\overrightarrow{60k}$
• D. None of these

Answer 1

Answer: (D)

None of these

$\overline{AB}=3i+4j-5k$
$\overline{BC}=-7i-j+5k$
$\overline{CA}=4i-3j$

Now

Unit vector along AB

$=\frac{\sqrt{2}}{10}[3i+4j-5k]$

Unit vector along BC

$=\frac{\sqrt{3}}{15}[-7i-j+5k]$

Unit vector along CA

$=\frac{1}{5}[4i-3j]$

Hence

$F_{AB}=\frac{1}{2}[9\sqrt{2}i+12\sqrt{2}j-15\sqrt{2}k]$

$F_{BC}=-7\sqrt{3}i-\sqrt{3}j+5\sqrt{3}k$
$F_{CA}=12i-9j$

Hence

$F_{AB}+F_{BC}+F_{AC}$
$=(\frac{9\sqrt{2}}{2}-7\sqrt{3}+12)i+j(6\sqrt{2}-\sqrt{3}-9)j+(5\sqrt{3}-\frac{15\sqrt{2}}{2})k$