List of top Mathematics Questions on Skew Lines

Represent the equations of lines $l_1$ and $l_2$ in vector form and check whether they are intersecting or not.
$l_1 : \frac{x+3}{-3} = \frac{y-1}{1} = \frac{z-5}{5}$
$l_2 : \frac{x+1}{-1} = \frac{2-y}{-2} = \frac{z-5}{5}$}

Find the vector equation of a line passing through the origin and perpendicular to both the lines \(\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k})\) and \(\vec{r} = \mu(\hat{i} - \hat{j} + \hat{k})\).

A line passing through the points \( A(1, 2, 3) \) and \( B(5, 8, 11) \) intersects the line \( \vec{r} = 4\hat{i} + \hat{j} + \lambda(5\hat{i} + 2\hat{j} + \hat{k}) \). Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.

Represent the equations of lines $l_1$ and $l_2$ in vector form and check whether they are intersecting or not. \[ l_1 : \frac{x + 3}{-3} = \frac{y - 1}{1} = \frac{z - 5}{5} \quad \text{and} \quad l_2 : \frac{x + 1}{-1} = \frac{2 - y}{-2} = \frac{z - 5}{5} \]

A line passing through the points A(1, 2, 3) and B(5, 8, 11) intersects the line \(\vec{r} = 4\hat{i} + \hat{j} + \lambda(5\hat{i} + 2\hat{j} + \hat{k})\). Find the co-ordinates of the point of intersection. Hence, write the vector equation of a line passing through the point of intersection and perpendicular to both the lines.

Find the shortest distance between the skew lines \[ \vec r=(1,0,0)+\lambda(1,1,0) \] and \[ \vec r=(0,1,1)+\mu(0,1,1) \]

Find the shortest distance between the lines \[ \vec r= \hat i+2\hat j+\hat k+\lambda(\hat i-\hat j+\hat k) \] and \[ \vec r= 2\hat i-\hat j-\hat k+\mu(2\hat i+\hat j+2\hat k) \]