List of top Mathematics Questions on Shortest Distance Between Skew Lines

The shortest distance between the lines \[ \vec r=\vec a+t\vec b \] and \[ \vec r=\vec c+s\vec d \] where \[ \vec a=\hat i-2\hat j+2\hat k, \quad \vec b=3\hat i-2\hat j-2\hat k, \] \[ \vec c=6\hat i+2\hat j+2\hat k, \quad \vec d=-4\hat i-\hat k, \] is

The shortest distance between the lines
\(\vec{r} = (\frac{1}{3}\hat{i} + \frac{8}{3}\hat{j} - \frac{1}{3}\hat{k}) + \lambda(2\hat{i} - 5\hat{j} + 6\hat{k})\)
and \(\vec{r} = (-\frac{2}{3}\hat{i} - \frac{1}{3}\hat{k}) + \mu(\hat{j} - \hat{k}), \lambda, \mu \in \mathbb{R}\), is:

The shortest distance between the lines \[ \frac{x-4}{1}=\frac{y-3}{2}=\frac{z-2}{-3} \] and \[ \frac{x+2}{2}=\frac{y-6}{4}=\frac{z-5}{-5} \] is :

The square of the distance of the point of intersection of the lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j})$, $a \neq 0$ and $\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k})$ from the origin is:

The shortest distance between the two lines \[ \vec r=(\hat i-\hat j)+s(\hat j+2\hat k) \] and \[ \vec r=(2\hat i+\hat k)+t(\hat i-\hat j+\hat k) \] is

The value of the integral \(\int_{0}^{2} \frac{\sqrt{x(x^2 + x + 1)}}{(\sqrt{x+1})(\sqrt{x^4 + x^2 + 1})} \, dx\) is equal to:

If the square of the shortest distance between the lines \[ \frac{x-2}{1} = \frac{y-1}{2} = \frac{z+3}{-3} \] and \[ \frac{x+1}{2} = \frac{y+3}{4} = \frac{z+5}{-5} \] is \( \frac{m}{n} \), where \( m \) and \( n \) are co-prime numbers, then \( m+n \) is equal to: