Question:
Let
\[
\overrightarrow{OA}=\hat{i}+2\hat{j}-4\hat{k}
\]
and
\[
\overrightarrow{OB}=3\hat{i}-4\hat{j}-2\hat{k}
\]
be the position vectors of points \(A\) and \(B\). If a point \(C\) divides the line segment \(AB\) in the ratio \(1:3\) externally, then the position vector of a point which divides \(OC\) in the ratio \(4:1\) internally is
Let
\[
\overrightarrow{OA}=\hat{i}+2\hat{j}-4\hat{k}
\]
and
\[
\overrightarrow{OB}=3\hat{i}-4\hat{j}-2\hat{k}
\]
be the position vectors of points \(A\) and \(B\). If a point \(C\) divides the line segment \(AB\) in the ratio \(1:3\) externally, then the position vector of a point which divides \(OC\) in the ratio \(4:1\) internally is
Show Hint
Remember the section formula:
\[
\text{External division: }
\frac{m\vec{b}-n\vec{a}}{m-n},
\qquad
\text{Internal division: }
\frac{m\vec{b}+n\vec{a}}{m+n}.
\]
Always find the external division point first and then apply the internal division formula if required.
Updated On: Jul 9, 2026
- \(5(\hat{i}-\hat{j})\)
- \(\hat{i}-4\hat{j}+2\hat{k}\)
- \(4\hat{i}-2\hat{j}+\hat{k}\)
- \(4(\hat{j}-\hat{k})\) \bigskip
Show Solution
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The Correct Option is D
Solution and Explanation
Concept:
If a point divides the line joining points with position vectors \(\vec{a}\) and \(\vec{b}\) externally in the ratio \(m:n\), then its position vector is
\[
\frac{m\vec{b}-n\vec{a}}{m-n}.
\]
If a point divides a line segment internally in the ratio \(m:n\), then its position vector is
\[
\frac{m\vec{b}+n\vec{a}}{m+n}.
\]
Step 1: Find the position vector of point \(C\). Given \[ \vec{A}=\hat{i}+2\hat{j}-4\hat{k}, \] \[ \vec{B}=3\hat{i}-4\hat{j}-2\hat{k}. \] Since \(C\) divides \(AB\) externally in the ratio \(1:3\), \[ \vec{OC} = \frac{1\vec{B}-3\vec{A}}{1-3}. \] \[ = \frac{(3\hat{i}-4\hat{j}-2\hat{k}) -3(\hat{i}+2\hat{j}-4\hat{k})}{-2}. \] \[ = \frac{3\hat{i}-4\hat{j}-2\hat{k} -3\hat{i}-6\hat{j}+12\hat{k}}{-2}. \] \[ = \frac{-10\hat{j}+10\hat{k}}{-2}. \] \[ = 5\hat{j}-5\hat{k}. \] Hence, \[ \vec{OC}=5(\hat{j}-\hat{k}). \]
Step 2: Find the point dividing \(OC\) internally in the ratio \(4:1\). Let the required point be \(P\). Since \(P\) divides \(OC\) internally in the ratio \(4:1\), \[ \vec{OP} = \frac{4\vec{OC}+1\cdot \vec{OO}}{4+1}. \] Since \[ \vec{OO}=\vec{0}, \] \[ \vec{OP} = \frac{4}{5}\vec{OC}. \] Substituting \(\vec{OC}=5(\hat{j}-\hat{k})\), \[ \vec{OP} = \frac{4}{5}\times 5(\hat{j}-\hat{k}). \] \[ = 4(\hat{j}-\hat{k}). \]
Step 3: Write the final answer. \[ \boxed{4(\hat{j}-\hat{k})} \]
Step 1: Find the position vector of point \(C\). Given \[ \vec{A}=\hat{i}+2\hat{j}-4\hat{k}, \] \[ \vec{B}=3\hat{i}-4\hat{j}-2\hat{k}. \] Since \(C\) divides \(AB\) externally in the ratio \(1:3\), \[ \vec{OC} = \frac{1\vec{B}-3\vec{A}}{1-3}. \] \[ = \frac{(3\hat{i}-4\hat{j}-2\hat{k}) -3(\hat{i}+2\hat{j}-4\hat{k})}{-2}. \] \[ = \frac{3\hat{i}-4\hat{j}-2\hat{k} -3\hat{i}-6\hat{j}+12\hat{k}}{-2}. \] \[ = \frac{-10\hat{j}+10\hat{k}}{-2}. \] \[ = 5\hat{j}-5\hat{k}. \] Hence, \[ \vec{OC}=5(\hat{j}-\hat{k}). \]
Step 2: Find the point dividing \(OC\) internally in the ratio \(4:1\). Let the required point be \(P\). Since \(P\) divides \(OC\) internally in the ratio \(4:1\), \[ \vec{OP} = \frac{4\vec{OC}+1\cdot \vec{OO}}{4+1}. \] Since \[ \vec{OO}=\vec{0}, \] \[ \vec{OP} = \frac{4}{5}\vec{OC}. \] Substituting \(\vec{OC}=5(\hat{j}-\hat{k})\), \[ \vec{OP} = \frac{4}{5}\times 5(\hat{j}-\hat{k}). \] \[ = 4(\hat{j}-\hat{k}). \]
Step 3: Write the final answer. \[ \boxed{4(\hat{j}-\hat{k})} \]
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