Question:
If the lines \(\overset{⃗}{r} = (\hat{i}+m\hat{j}+3\hat{k})+λ(2\hat{i}+3\hat{j}+4\hat{k})\) and \(\overset{⃗}{r} = (4\hat{i}+\hat{j})+μ(5\hat{i}+m\hat{j}+\hat{k})\) intersect each other, then m =
If the lines \(\overset{⃗}{r} = (\hat{i}+m\hat{j}+3\hat{k})+λ(2\hat{i}+3\hat{j}+4\hat{k})\) and \(\overset{⃗}{r} = (4\hat{i}+\hat{j})+μ(5\hat{i}+m\hat{j}+\hat{k})\) intersect each other, then m =
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Equate the position vectors of the two lines and solve for the parameters and m.
Updated On: Oct 1, 2026
- \(1\)
- \(-1\)
- \(2\)
- \(-2\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept
Two lines intersect when there are values of \(\lambda\) and \(\mu\) that give the same point.
Step 2: Equate coordinates
\(1 + 2\lambda = 4 + 5\mu\) ... (i)
\(m + 3\lambda = 1 + m\mu\) ... (ii)
\(3 + 4\lambda = \mu\) ... (iii)
Put (iii) into (i): \(1 + 2\lambda = 4 + 15 + 20\lambda\), so \(-18\lambda = 18\) and \(\lambda = -1\). Then \(\mu = 3 - 4 = -1\).
Step 3: Use (ii)
\[ m - 3 = 1 - m \Rightarrow 2m = 4 \Rightarrow m = 2 \]
So \(m = 2\), option (C).
Final Answer:
\(m = 2\), option (C). \[ \boxed{2} \]
Two lines intersect when there are values of \(\lambda\) and \(\mu\) that give the same point.
Step 2: Equate coordinates
\(1 + 2\lambda = 4 + 5\mu\) ... (i)
\(m + 3\lambda = 1 + m\mu\) ... (ii)
\(3 + 4\lambda = \mu\) ... (iii)
Put (iii) into (i): \(1 + 2\lambda = 4 + 15 + 20\lambda\), so \(-18\lambda = 18\) and \(\lambda = -1\). Then \(\mu = 3 - 4 = -1\).
Step 3: Use (ii)
\[ m - 3 = 1 - m \Rightarrow 2m = 4 \Rightarrow m = 2 \]
So \(m = 2\), option (C).
Final Answer:
\(m = 2\), option (C). \[ \boxed{2} \]
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