If P , Q are two points on the curve y = 2 x + 2 in the rectangular Cartesian coordinate system such that O…

Given,

${\overline{OP}}_{.}.\hat{i}=-1\Rightarrow P_x=-1$

$P$ lies on $y=2^{x+2}$

$\Rightarrow y=2^{-1+2}$

$\Rightarrow y=2$

So, $\left(P_x,P_y\right)=\left(-1,2\right)$

Now $\overline{OQ}·\hat{i}=2\Rightarrow Q_x=2$

Also $Q$ lies on $y=2^{x+2}$

$\Rightarrow y=2^{2+2}$

$\Rightarrow y=16$

So, $\left(Q_x,Q_y\right)=\left(2,16\right)$

Now $\overline{OQ}=2i+16j$

And $\overline{OP}=-i+2j$

So, $\overline{OQ}-4\overline{OP}$$=2i+16j+4i-8j$$=6i+8j$