Vector Algebra
Vector Algebra
nta_abhyas_2025
Grade 12
Question:
If two points $P$ and $Q$ are on the curve $y = 2^{x+1}$, such that $\overrightarrow{OP} \cdot \vec{i} = -1$ and $\overrightarrow{OQ} \cdot \vec{i} = 2$, where $\vec{i}$ is a unit vector along the $x$-axis, then $|\overrightarrow{OQ} - \overrightarrow{OP}|$ is equal to
Step-by-Step Solution
Key Concept: Vector difference magnitude is computed using the distance formula after expressing position vectors on a parametric curve.
Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two points on the curve $y = 2^{x+2}$. Then $\vec{OP} = -\vec{i} + 2z_1 = -\vec{i}$ and $z_1 = 1$, so $y_1 = 2$. Also $\vec{OQ} = -2\vec{i} + x_2 = 2$ and $y_0 = 16$, giving $\vec{OP} = -\vec{i} + 2j$ and $\vec{OQ} = -2\vec{i} + 16\vec{j}$. Then $\vec{OQ} - 4\vec{OP} = -6\vec{i} + 8\vec{j}$ and $|\vec{OQ} - 4\vec{OP}| = \sqrt{36 + 64} = \sqrt{100} = 10$.
Correct Answer: 10