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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Recognize Pythagorean triplets to save time! \( (7, 24, 25) \) is a standard triplet, meaning \( 7^2 + 24^2 = 25^2 \). Always include both positive and negative roots when taking a square root for a coordinate.
Updated On: Sep 10, 2026
- \( \pm 49 \)
- \( \pm 5 \)
- \( \pm 1 \)
- \( \pm 7 \)
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The Correct Option is D
Solution and Explanation
Concept:
• The position vector of a point \( (x, y) \) is \( \vec{p} = x\hat{i} + y\hat{j} \).
• The magnitude (length) of a vector \( \vec{v} = a\hat{i} + b\hat{j} \) is given by \( |\vec{v}| = \sqrt{a^2 + b^2} \).
Step 1: Express the position vector and its magnitude
For the point \( (24, n) \), the position vector is:
\[ \vec{p} = 24\hat{i} + n\hat{j} \]
The magnitude is given as:
\[ |\vec{p}| = \sqrt{24^2 + n^2} \]
Step 2: Set up the equation using the given magnitude
We are given that \( |\vec{p}| = 25 \).
\[ \sqrt{24^2 + n^2} = 25 \]
Square both sides of the equation to remove the square root:
\[ 24^2 + n^2 = 25^2 \]
Step 3: Calculate numerical values and solve for \( n \)
Calculate the squares of 24 and 25:
\[ 576 + n^2 = 625 \]
Isolate \( n^2 \):
\[ n^2 = 625 - 576 \]
\[ n^2 = 49 \]
Take the square root of both sides:
\[ n = \pm \sqrt{49} = \pm 7 \]
• The position vector of a point \( (x, y) \) is \( \vec{p} = x\hat{i} + y\hat{j} \).
• The magnitude (length) of a vector \( \vec{v} = a\hat{i} + b\hat{j} \) is given by \( |\vec{v}| = \sqrt{a^2 + b^2} \).
Step 1: Express the position vector and its magnitude
For the point \( (24, n) \), the position vector is:
\[ \vec{p} = 24\hat{i} + n\hat{j} \]
The magnitude is given as:
\[ |\vec{p}| = \sqrt{24^2 + n^2} \]
Step 2: Set up the equation using the given magnitude
We are given that \( |\vec{p}| = 25 \).
\[ \sqrt{24^2 + n^2} = 25 \]
Square both sides of the equation to remove the square root:
\[ 24^2 + n^2 = 25^2 \]
Step 3: Calculate numerical values and solve for \( n \)
Calculate the squares of 24 and 25:
\[ 576 + n^2 = 625 \]
Isolate \( n^2 \):
\[ n^2 = 625 - 576 \]
\[ n^2 = 49 \]
Take the square root of both sides:
\[ n = \pm \sqrt{49} = \pm 7 \]
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