A point $A$ has the coordinate $(5,4,0)$.

A line $l$ has the equation:

$r=i+6k+\lambda (2i+j+k)$ 

$P$ is the point on $l$ which is closest to $A$.
Determine the position vector of of $P$.

Find the shortest distance between the line  $\mathbf{\text{r}}=\left(\begin{array}{c}-5\\ 2\\ 1\end{array}\right)+\lambda \left(\begin{array}{c}1\\ 4\\ 2\end{array}\right)$   and the point $(2,-3,1)$   

Give your answer correct to 2 decimal places.

Find the shortest distance between the line  $\mathbf{\text{r}}=\left(\begin{array}{c}4\\ -3\\ 2\end{array}\right)+\lambda \left(\begin{array}{c}2\\ 1\\ -4\end{array}\right)$   and the point $(1,4,1)$   

Give your answer correct to 2 decimal places.

Find the shortest distance between the two parallel lines  $\mathbf{\text{r}}=\left(\begin{array}{c}6\\ 2\\ 5\end{array}\right)+\lambda \left(\begin{array}{c}-4\\ 3\\ 1\end{array}\right)$   and    $\mathbf{\text{r}}=\left(\begin{array}{c}1\\ 2\\ 1\end{array}\right)+\mu \left(\begin{array}{c}-4\\ 3\\ 1\end{array}\right)$   

Give your answer correct to 2 decimal places.

Find the shortest distance between the two parallel lines  $\mathbf{\text{r}}=\left(\begin{array}{c}3\\ 1\\ 2\end{array}\right)+\lambda \left(\begin{array}{c}6\\ 5\\ -2\end{array}\right)$   and    $\mathbf{\text{r}}=\left(\begin{array}{c}5\\ 3\\ 6\end{array}\right)+\mu \left(\begin{array}{c}6\\ 5\\ -2\end{array}\right)$   

Give your answer correct to 2 decimal places.

Find the shortest distance between the line  $\mathbf{\text{r}}=\left(\begin{array}{c}-4\\ 2\\ 1\end{array}\right)+\lambda \left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$   and the point $(3,1,1)$   

Give your answer correct to 2 decimal places.

Find the shortest distance between the line  $\mathbf{\text{r}}=\left(\begin{array}{c}2\\ -1\\ 4\end{array}\right)+\lambda \left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)$   and the point $(2,2,1)$   

Give your answer correct to 2 decimal places.

Find the shortest distance between the two parallel lines  $\mathbf{\text{r}}=\left(\begin{array}{c}5\\ 1\\ 2\end{array}\right)+\lambda \left(\begin{array}{c}-3\\ -3\\ 2\end{array}\right)$   and    $\mathbf{\text{r}}=\left(\begin{array}{c}1\\ 4\\ 3\end{array}\right)+\mu \left(\begin{array}{c}-3\\ -3\\ 2\end{array}\right)$   

Give your answer correct to 2 decimal places.

Find the shortest distance between the two parallel lines  $\mathbf{\text{r}}=\left(\begin{array}{c}2\\ 6\\ 1\end{array}\right)+\lambda \left(\begin{array}{c}5\\ 4\\ -3\end{array}\right)$   and    $\mathbf{\text{r}}=\left(\begin{array}{c}3\\ 1\\ 2\end{array}\right)+\mu \left(\begin{array}{c}5\\ 4\\ -3\end{array}\right)$   

Give your answer correct to 2 decimal places.

[Edexcel FP3 June 2009 Q7c Edited]

The lines $l_1$ and $l_2$ have equations

$\mathbf{\text{r}}=\left(\begin{array}{c}1\\ -1\\ 2\end{array}\right)+\lambda \left(\begin{array}{c}-1\\ 3\\ 4\end{array}\right)$ and $\mathbf{\text{r}}=\left(\begin{array}{c}\alpha \\ -4\\ 0\end{array}\right)+\mu \left(\begin{array}{c}0\\ 3\\ 2\end{array}\right)$

When $\alpha \ne 1$, the lines $l_1$ and $l_2$ do not intersect and are skew lines.

Given that $\alpha =2$, find the shortest distance between the lines $l_1$ and $l_2$.

(3 marks)

[Edexcel FP3 June 2018 Q6b]

The line $l_1$ has equation

$$r=i+2k+\lambda (2i+3j-k)$$

where $\lambda$ is a scalar parameter.

The line $l_2$ has equation

$$\frac{x+1}{1}=\frac{y-4}{1}=\frac{z-1}{3}$$

Find the shortest distance between the lines $l_1$ and $l_2$.

(5 marks)

[OCR C4 June 2011 Q5iii Edited]

The lines $l_1$ and $l_2$ have equation

$r=\left(\begin{array}{c}4\\ 6\\ 4\end{array}\right)+s\left(\begin{array}{c}3\\ 2\\ 1\end{array}\right)$        and       $r=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)+t\left(\begin{array}{c}0\\ 1\\ -1\end{array}\right)$ 

respectively.

It can be shown that the line are skew.

The point $A$ lies on $l_1$ and $OA$ is perpendicular to $l_1$, where $O$ is the origin. Find the position vector of $A$.

(3 marks)