Analytical Space Geometry

9000101005

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

9000101006

Level: 
A
Find the value of the real parameter \(m\) which ensures that the following lines are parallel and not identical lines. \[ \begin{aligned}p\colon x& = 1 + t, & \\y & = 2 - t, \\z & = 1 - t,\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = s, & \\y & = 1 + s, \\z & = 3 + ms,\ s\in \mathbb{R} \\ \end{aligned} \]
No solution exists.
The lines are parallel and not identical for every real \(m\).
\(m = -2\)
\(m = 2\)

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 \(m\).
\(m = -2\)
\(m = 2\)

9000101009

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[\begin{aligned} a\colon x & = t, & & \\y & = -t, & & \\z & = 1 - t,\ t\in \mathbb{R} & & \end{aligned}\]\[\begin{aligned} b\colon x & = -s, & & \\y & = s, & & \\z & = 1 + s,\ s\in \mathbb{R} & & \end{aligned}\]
identical lines
skew lines
intersecting lines
parallel, not identical lines

9000101010

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[\begin{aligned} a\colon x & = t, & & \\y & = -t, & & \\z & = 1 - t,\ t\in \mathbb{R} & & \end{aligned}\]\[\begin{aligned} b\colon x & = -s, & & \\y & = s, & & \\z & = -1 + s,\ s\in \mathbb{R} & & \end{aligned}\]
parallel, not identical lines
skew lines
intersecting lines
identical lines

9000101103

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