Analytic geometry in a space

9000101004

Level: 
A
Find all the values of the real parameter \(m\) so that the lines \(p\) and \(q\) are skew lines. \[ \begin{aligned}p\colon x& = 1 + t, & \\y & = 2 - t, \\z & = 1 - t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = s, & \\y & = 1 + s, \\z & = 3 + ms;\ s\in \mathbb{R} \\ \end{aligned} \]
\(m\in\mathbb{R}\setminus\{-2\}\)
No solution exists.
The lines are skew for every real \(m\).
\(m = -2\)

9000101007

Level: 
A
Find the value of the real parameter \(m\) which ensures that the following two lines are identical. \[ \begin{aligned}p\colon x& = 1 + t, & \\y & = 2 - t, \\z & = 1 - t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = s, & \\y & = 1 + s, \\z & = 3 + ms;\ s\in \mathbb{R} \\ \end{aligned} \]
No solution exists.
The lines are identical for every real \(m\).
\(m = -2\)
\(m = 2\)

9000101103

Level: 
B
Find the distance between two parallel lines \(p\) and \(q\). \[ \begin{aligned}p\colon x& = 2, & \\y & = 3t, \\z & = 1 - t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = 3, & \\y & = 6s, \\z & = 1 - 2s;\ s\in \mathbb{R} \\ \end{aligned} \]
\(1\)
\(2\)
\(3\)
\(4\)