If position vector \( \vec{p} \) of a point \( (24, n) \) is such that \( |\vec{p}| = 25 \), then the value of \( n \) is :
Show Hint
Pythagorean triplets can speed up calculations: \( 7, 24, 25 \) is a common triplet.
- \( \pm 49 \)
- \( \pm 5 \)
- \( \pm 1 \)
- \( \pm 7 \)
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 of a vector \(\vec{v}=a\hat{i}+b\hat{j}\) is: \[ |\vec{v}|=\sqrt{a^2+b^2} \]
Step 1: Write the position vector
Given the point \((24,n)\), its position vector is: \[ \vec{p}=24\hat{i}+n\hat{j} \]
Step 2: Apply the magnitude formula
Given: \[ |\vec{p}|=25 \] Therefore, \[ \sqrt{24^2+n^2}=25 \]
Step 3: Solve for \(n\)
Squaring both sides: \[ 24^2+n^2=25^2 \] \[ 576+n^2=625 \] \[ n^2=625-576 \] \[ n^2=49 \] Taking the square root: \[ n=\pm\sqrt{49} \] \[ n=\pm 7 \]
Final Answer:
Therefore, \[ \boxed{n=\pm 7} \] Hence, the correct answer is Option (D).
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