Question:
If position vector \( \vec{p} \) of a point \( (24, n) \) is such that \( |\vec{p}| = 25 \), then the value of \( n \) is :
If position vector \( \vec{p} \) of a point \( (24, n) \) is such that \( |\vec{p}| = 25 \), then the value of \( n \) is :
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This problem uses the well-known Pythagorean triple \( (7, 24, 25) \). Remembering basic right-triangle integer triples allows you to verify vector components instantly.
Updated On: Jul 6, 2026
- \( \pm 49 \)
- \( \pm 5 \)
- \( \pm 1 \)
- \( \pm 7 \)
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The Correct Option is D
Solution and Explanation
Concept:
The magnitude of a 2D position vector \( \vec{p} = x\hat{i} + y\hat{j} \) corresponding to a coordinate point \( (x, y) \) is calculated using the Pythagorean-based distance formula:
\[
|\vec{p}| = \sqrt{x^2 + y^2}
\]
Step 1: Write the position vector format and vector magnitude. The coordinate point is given as \( (24, n) \), so its vector representation is: \[ \vec{p} = 24\hat{i} + n\hat{j} \] The magnitude formula yields: \[ |\vec{p}| = \sqrt{24^2 + n^2} \]
Step 2: Form an equation with the given magnitude. We are given that \( |\vec{p}| = 25 \), so: \[ \sqrt{24^2 + n^2} = 25 \] Square both sides of the equation to eliminate the radical sign: \[ 24^2 + n^2 = 25^2 \]
Step 3: Solve for parameter \( n \). Compute the known squares: \[ 576 + n^2 = 625 \] Isolate \( n^2 \): \[ n^2 = 625 - 576 \implies n^2 = 49 \] Taking square roots on both sides gives both positive and negative options: \[ n = \pm\sqrt{49} = \pm 7 \]
Step 1: Write the position vector format and vector magnitude. The coordinate point is given as \( (24, n) \), so its vector representation is: \[ \vec{p} = 24\hat{i} + n\hat{j} \] The magnitude formula yields: \[ |\vec{p}| = \sqrt{24^2 + n^2} \]
Step 2: Form an equation with the given magnitude. We are given that \( |\vec{p}| = 25 \), so: \[ \sqrt{24^2 + n^2} = 25 \] Square both sides of the equation to eliminate the radical sign: \[ 24^2 + n^2 = 25^2 \]
Step 3: Solve for parameter \( n \). Compute the known squares: \[ 576 + n^2 = 625 \] Isolate \( n^2 \): \[ n^2 = 625 - 576 \implies n^2 = 49 \] Taking square roots on both sides gives both positive and negative options: \[ n = \pm\sqrt{49} = \pm 7 \]
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