Question

In the cartesian plane, $O$ is the origin of the coordinate axes. A person starts at $O$ and walks a distance of $3$ units in the NORTH-EAST direction and reaches the point $P.$ From $P$ he walks $4$ units distance parallel to NORTH-WEST direction and reaches the point $Q.$ Express the vector $\overline{OQ}$ in terms of $\hat{i}$ and $\hat{j}.$

Answer 1

Hint: North-east is halfway between north and east. So, the angle it makes with the $x-$axis is $45{}^\circ .$ North-west is halfway between north and west. So, the angle it makes with the $y-$ axis is $\left( 90+45 \right){}^\circ =135{}^\circ .$

Complete step by step solution:
It is given that the person walks a distance of $3$ units in the north-east direction and reaches the point $P.$
Since we are discussing the direction, we need to consider both the $x-$component and $y-$component of the direction. We know that the angle made by the path in which the person walks and the $x-$axis is $45{}^\circ .$
Thus, we will get the vector $\overline{OP}=3\cos 45{}^\circ \hat{i}+3\sin 45{}^\circ \hat{j}.$
And this gives us $\overline{OP}=\dfrac{3}{\sqrt{2}}\hat{i}+\dfrac{3}{\sqrt{2}}\hat{j}.$
Also, given that the person walks $4$ units from $P$ in the north-west direction and reaches $Q.$
Here also we are discussing the direction in which the person walks. And so, we are supposed to consider both the $x-$component and the $y-$component of the direction. The angle made by the path in which the person walks from $P$ to $Q$ and the $x-$axis is $135{}^\circ .$
Therefore, we will get $\overline{PQ}=4\cos 135{}^\circ \hat{i}+4\sin 135{}^\circ \hat{j}.$
And from this, $\overline{PQ}=-\dfrac{4}{\sqrt{2}}\hat{i}+\dfrac{4}{\sqrt{2}}\hat{j}.$
And now, as per the question, we need to find the vector $\overline{OQ}$ in terms of $\hat{i}$ and $\hat{j},$ and so, we add $\overline{OP}$ and $\overline{PQ}$
That is, $\overline{OQ}=\overline{OP}+\overline{PQ}=\dfrac{3}{\sqrt{2}}\hat{i}+\dfrac{3}{\sqrt{2}}\hat{j}-\dfrac{4}{\sqrt{2}}\hat{i}+\dfrac{4}{\sqrt{2}}\hat{j}=-\dfrac{1}{\sqrt{2}}\hat{i}+\dfrac{7}{\sqrt{2}}\hat{j}.$
Hence $\overline{OQ}=\overline{OP}+\overline{PQ}=\dfrac{1}{\sqrt{2}}\left( \hat{i}+7\hat{j} \right).$

Note: The half way between the cardinal directions are collectively called the intercardinal directions. And they are north-east, north-west, south-east and south-west. The angles they make with the $x-$axis are $NE=45{}^\circ , NW=135{}^\circ , SW=225{}^\circ , SE=315{}^\circ .$ Also, remember that $\sin 45{}^\circ =\cos 45{}^\circ =\dfrac{1}{\sqrt{2}}, $ $\cos 135{}^\circ =\cos \left( 180-45 \right){}^\circ =-\dfrac{1}{\sqrt{2}}, \sin 135{}^\circ =\sin \left( 180-45 \right){}^\circ =\dfrac{1}{\sqrt{2}}.$