Two given points $P$ and $Q$ in the rectangular cartesian coordinates lie on $y = 2^{x^2}$ such that $\overrightarrow{OP} \cdot \hat{i} = -1$ and $\overrightarrow{OQ} \cdot \hat{i} = +2$ where $\hat{i}$ is a unit vector along the x-axis. The magnitude of $\frac{\overrightarrow{OQ} - 4\overrightarrow{OP}}{2}$ is _______.
Step-by-Step Solution
Key Concept: Convert points on a curve to position vectors and use vector arithmetic to find the required magnitude.
Given two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ on the curve $y = 2^{x+2}$, the projections onto the $x$-axis are $x_1 = -1$ giving $y_1 = 2$ and $x_2 = 2$ giving $y_2 = 16$. With unit vector $\vec{j}$ along the $y$-axis, we have $\vec{OP} = -\vec{i} + 2\vec{j}$ and $\vec{OQ} = 2\vec{i} + 16\vec{j}$. The vector $\vec{OQ} - 4\vec{OP} = 6\vec{i} + 8\vec{j}$ has magnitude $|\vec{OQ} - 4\vec{OP}| = \sqrt{36 + 64} = 10$.
Correct Answer: Let me verify the solution step by step.
**Given information:**
- Points P and Q lie on $y = 2^{x^2}$
- $\overrightarrow{OP} \cdot \hat{i} = -1$ (x-coordinate of P is -1)
- $\overrightarrow{O