Question:
The ratio in which the point \(\hat{i}+2\hat{j}+3\hat{k}\) divides the line segment joining \(-2\hat{i}+3\hat{j}+5\hat{k}\) and \(7\hat{i}-\hat{k}\) is....
The ratio in which the point \(\hat{i}+2\hat{j}+3\hat{k}\) divides the line segment joining \(-2\hat{i}+3\hat{j}+5\hat{k}\) and \(7\hat{i}-\hat{k}\) is....
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Use the section formula and compare one coordinate, then verify with the others.
Updated On: Nov 12, 2026
- \(1:2\)
- \(2:3\)
- \(3:4\)
- \(1:4\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
Let the point divide the segment in the ratio \(m:n\). The section formula gives \(\dfrac{m\bar b+n\bar a}{m+n}\), with \(\bar a=-2\hat i+3\hat j+5\hat k\) and \(\bar b=7\hat i-\hat k\).
Step 2: Use the i coefficient
\(\dfrac{7m-2n}{m+n}=1\Rightarrow 6m=3n\Rightarrow n=2m\).
Step 3: Ratio
So \(m:n=1:2\).
Step 4: Verify
The \(\hat j\) coefficient is \(\dfrac{3n}{m+n}=\dfrac{6}{3}=2\) and the \(\hat k\) coefficient is \(\dfrac{-1+10}{3}=3\), both match.
Final Answer:
The point divides the segment in the ratio 1:2. \[ \boxed{\text{(A) }1:2} \]
Let the point divide the segment in the ratio \(m:n\). The section formula gives \(\dfrac{m\bar b+n\bar a}{m+n}\), with \(\bar a=-2\hat i+3\hat j+5\hat k\) and \(\bar b=7\hat i-\hat k\).
Step 2: Use the i coefficient
\(\dfrac{7m-2n}{m+n}=1\Rightarrow 6m=3n\Rightarrow n=2m\).
Step 3: Ratio
So \(m:n=1:2\).
Step 4: Verify
The \(\hat j\) coefficient is \(\dfrac{3n}{m+n}=\dfrac{6}{3}=2\) and the \(\hat k\) coefficient is \(\dfrac{-1+10}{3}=3\), both match.
Final Answer:
The point divides the segment in the ratio 1:2. \[ \boxed{\text{(A) }1:2} \]
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