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# A man walks 4 mtrs. towards west and then walks 3 mtrs in the direction of $\large\frac{\pi}{6}\:$ radians east of north. What is the displacement of the man from the starting point?

$\begin{array}{1 1} (A) -\large\frac{11}{4}\hat i+\large\frac{3\sqrt 3}{4}\hat j \\ (B) -\large\frac{11}{2}\hat i+\large\frac{3\sqrt 3}{2}\hat j \\ (C) -\large\frac{11}{2}\hat i+\large\frac{27}{4}\hat j \\ (D) -\large\frac{11}{4}\hat i+\large\frac{3}{2}\hat j\end{array}$

Let the man start from origin. He walks west side 4 metrs.
$\therefore\:\overrightarrow {OA}=-4\hat i$
Then he walks 3 metrs. $\large\frac{\pi}{6}$ east of north and moves to $B$
$\therefore \: |\overrightarrow {AB}|=3$, $< BAO=\large\frac{\pi}{3}$ and $|\overrightarrow {OA}|=4$
Let $\overrightarrow {OB}=x\hat i+y\hat j$
From triangular law of addition $\overrightarrow {OA}+\overrightarrow {AB}=\overrightarrow {OB}$
$\Rightarrow\:-4\hat i+\overrightarrow {AB}=x\hat i+y\hat k$
$\Rightarrow\:\overrightarrow {AB}=x\hat i+y\hat k+4\hat i=(x+4)\hat i+y\hat k$
Also $cos\large\frac{\pi}{3}$$=\large\frac{\overrightarrow {OA}.\overrightarrow {AB}}{|\overrightarrow {OA}||\overrightarrow {AB}|}=\frac{(-4\hat i).((x+4)\hat i+y\hat j)}{12} \Rightarrow\:\large\frac{1}{2}=\frac{-4x-16}{12} \Rightarrow\:x=-\large\frac{11}{2} Now |\overrightarrow {AB}|=3\Rightarrow\:(x+4)^2+y^2=9 Substituting the value of x we get y^2=9-\large\frac{9}{4} \Rightarrow\:y=\large\frac{3\sqrt 3}{2} \therefore The required displacement \overrightarrow {OB}=-\large\frac{11}{2}$$\hat i+\large\frac{3\sqrt 3}{2}$$\hat j$
edited Dec 5, 2013 by pady_1