List of top Mathematics Questions on Distance between Two Lines asked in CBSE CLASS XII

Find the value of \( p \) if the shortest distance between the lines \( \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( \vec{r} = (p\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k}) \) is \( \frac{3}{\sqrt{2}} \) units.

Find the value of \( p \) if the shortest distance between the lines \( \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( \vec{r} = (p\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k}) \) is \( \frac{3}{\sqrt{2}} \) units.

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

Find:

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

Find the shortest distance between the lines: \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}, \quad \frac{x + 2}{1} = \frac{y - 3}{-2} = \frac{z + 1}{2} \]