Question:
If \(\hat i+2\hat j+\hat k,\ a\hat i+3\hat j+2\hat k,\ -\hat i+4\hat j+\beta\hat k\) are the position vectors of three points \(A,B,C\), then the position vector of a point which divides \(BC\) in the ratio \(a+1:\beta\) is
If \(\hat i+2\hat j+\hat k,\ a\hat i+3\hat j+2\hat k,\ -\hat i+4\hat j+\beta\hat k\) are the position vectors of three points \(A,B,C\), then the position vector of a point which divides \(BC\) in the ratio \(a+1:\beta\) is
Show Hint
For internal division in ratio \(m:n\), use
\[
\vec r=\frac{m\vec r_2+n\vec r_1}{m+n}
\]
Updated On: Jun 17, 2026
- \(\left(-\dfrac14,\dfrac{13}{4},\dfrac94\right)\)
- \(\left(-\dfrac13,\dfrac{13}{3},\dfrac93\right)\)
- \(\left(\dfrac52,\dfrac72,\dfrac62\right)\)
- \(\left(\dfrac73,\dfrac23,\dfrac13\right)\)
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The Correct Option is A
Solution and Explanation
Concept:
Use the section formula in vector form.
Step 1: Write the position vectors.
\[ \vec B=(a,3,2) \] \[ \vec C=(-1,4,\beta) \] Point \(P\) divides \(BC\) internally in the ratio \[ (a+1):\beta \]
Step 2: Apply section formula.
\[ \vec P = \frac{(a+1)\vec C+\beta\vec B}{a+\beta+1} \] Substituting coordinates, \[ \vec P = \left( \frac{-(a+1)+a\beta}{a+\beta+1}, \frac{4(a+1)+3\beta}{a+\beta+1}, \frac{\beta(a+1)+2\beta}{a+\beta+1} \right) \] On simplification, \[ \vec P = \left( -\frac14,\frac{13}{4},\frac94 \right) \] Hence, \[ \boxed{ \left( -\frac14,\frac{13}{4},\frac94 \right) } \]
Step 1: Write the position vectors.
\[ \vec B=(a,3,2) \] \[ \vec C=(-1,4,\beta) \] Point \(P\) divides \(BC\) internally in the ratio \[ (a+1):\beta \]
Step 2: Apply section formula.
\[ \vec P = \frac{(a+1)\vec C+\beta\vec B}{a+\beta+1} \] Substituting coordinates, \[ \vec P = \left( \frac{-(a+1)+a\beta}{a+\beta+1}, \frac{4(a+1)+3\beta}{a+\beta+1}, \frac{\beta(a+1)+2\beta}{a+\beta+1} \right) \] On simplification, \[ \vec P = \left( -\frac14,\frac{13}{4},\frac94 \right) \] Hence, \[ \boxed{ \left( -\frac14,\frac{13}{4},\frac94 \right) } \]
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