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The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

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If direction vectors are proportional, lines are parallel. If the scalar triple product equals zero, lines intersect and the shortest distance is exactly 0.

Updated On: Jul 8, 2026

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Top Questions on Distance between Two Lines

If the shortest distance between the lines \(\frac{x - \lambda}{-2} = \frac{y - 2}{1} = \frac{z - 1}{1}\) and \(\frac{x - \sqrt{3}}{1} = \frac{y - 1}{-2} = \frac{z - 2}{1}\) is 1, then the sum of all possible values of \( \lambda \) is:
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Let the line of the shortest distance between the lines \(L_1: \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})\)and \(L_2: \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(\hat{i} + \hat{j} - \hat{k})\) intersect \(L_1\) and \(L_2\) at \(P\) and \(Q\), respectively. If \((\alpha, \beta, \gamma)\) is the midpoint of the line segment \(PQ\), then \(2(\alpha + \beta + \gamma)\) is equal to \(\_\_\_\_\).
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If the shortest distance between the lines \[ \frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4} \quad \text{and} \quad \frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2}\] is \(\frac{38}{3\sqrt{5}} k\) and \[\int_{0}^{k} \lfloor x^2 \rfloor dx = \alpha - \sqrt{\alpha}, \]where \([x]\) denotes the greatest integer function, then \( 6\alpha^3 \) is equal to _____
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The shortest distance between the line:
\[ \frac{x-3}{4} = \frac{y+7}{-11} = \frac{z-1}{5} \] and \[ \frac{x-5}{3} = \frac{y-9}{-6} = \frac{z+2}{1} \] is:
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Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

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Solution and Explanation

Top Questions on Distance between Two Lines

Questions Asked in CBSE Class XII exam

Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]