If P and Q are two points on the curve y = 2^ | vedclass

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(C) Given that $P$ and $Q$ lie on the curve $y = 2^{x+2}$.For point $P$,$\overline{OP} \cdot \hat{i} = -1$,which implies the $x$-coordinate of $P$ is

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$6\hat{i} + 8\hat{j}$

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(C) Given that $P$ and $Q$ lie on the curve $y = 2^{x+2}$.For point $P$,$\overline{OP} \cdot \hat{i} = -1$,which implies the $x$-coordinate of $P$ is $x_P = -1$.Substituting $x_P = -1$ into the curve equation: $y_P = 2^{-1+2} = 2^1 = 2$.Thus,$P = (-1, 2)$ and $\overline{OP} = -\hat{i} + 2\hat{j}$.For point $Q$,$\overline{OQ} \cdot \hat{i} = 2$,which implies the $x$-coordinate of $Q$ is $x_Q = 2$.Substituting $x_Q = 2$ into the curve equation: $y_Q = 2^{2+2} = 2^4 = 16$.Thus,$Q = (2, 16)$ and $\overline{OQ} = 2\hat{i} + 16\hat{j}$.Now,calculate $\overline{OQ} - 4\overline{OP}$:$\overline{OQ} - 4\overline{OP} = (2\hat{i} + 16\hat{j}) - 4(-\hat{i} + 2\hat{j})$$= 2\hat{i} + 16\hat{j} + 4\hat{i} - 8\hat{j}$$= (2+4)\hat{i} + (16-8)\hat{j}$$= 6\hat{i} + 8\hat{j}$.

If $P$ and $Q$ are two points on the curve $y = 2^{x+2}$ in the rectangular Cartesian coordinate system such that $\overline{OP} \cdot \hat{i} = -1$ and $\overline{OQ} \cdot \hat{i} = 2$,then $\overline{OQ} - 4\overline{OP} = $

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