Space geometry

9000111807

Level: 
B
In the following list identify a line such that the angle between this line and the plane \[ 2x - y + 3z - 5 = 0 \] is \(30^{\circ }\).
\(\begin{aligned}[t] p\colon x& = 2 + t, & \\y & = 1 + 3t, \\z & = -2t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] r\colon x& = -2t, & \\y & = -3 + t, \\z & = 1 - 3t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 2 + 3t, & \\y & = 3 - 2t, \\z & = 3 + t;\ t\in \mathbb{R} \\ \end{aligned}\)

9000111804

Level: 
B
In the following list identify a line such that the line is parallel to \(s\) and the distance between both lines is \(\sqrt{5}\). \[ \begin{aligned}[t] s\colon x& = -1 + t,& \\y & = 2t, \\z & = 2 - t;\ t\in \mathbb{R} \\ \end{aligned} \]
\(\begin{aligned}[t] r\colon x& = 3 - 2t,& \\y & = 3 - 4t, \\z & = 2t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 1, & \\y & = -1 + 5t, \\z & = 2 - 2t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = -5 - t,& \\y & = 2 - 2t, \\z & = 2 + t;\ t\in \mathbb{R} \\ \end{aligned}\)

9000111805

Level: 
B
In the following list identify a plane which is parallel to the plane \(\delta \) and the distance between both planes is \(2\). \[ \delta \colon x - 2y + 2y - 2 = 0 \]
\(\begin{aligned}[t] \beta \colon x& = -4 + 2s, & \\y& = 1 + r + s, \\z& = 1 + r;\ r,s\in \mathbb{R} \\ \end{aligned}\)
\(\gamma \colon - x + 2y - 2z - 2 = 0\)
\(\alpha \colon 2x - 4y + z - 4 = 0\)

9000111806

Level: 
B
In the following list identify the line such that the angle between this line and the line \(s\) is \(60^{\circ }\). \[ \begin{aligned}[t] s\colon x& = 2 + t, & \\y & = -1 - 2t, \\z & = 3 - t;\ t\in \mathbb{R} \\ \end{aligned} \]
\(\begin{aligned}[t] r\colon x& = t, & \\y & = -3 + t, \\z & = 1 + 2t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 1, & \\y & = -1 - t, \\z & = 3 + 2t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = -5 - 2t,& \\y & = 2 + 4t, \\z & = 2 + 2t;\ t\in \mathbb{R} \\ \end{aligned}\)

9000111808

Level: 
B
In the following list identify a plane such that the angle between this plane and the plane \(\rho \) is \(45^{\circ }\). \[ \rho \colon \begin{aligned}[t] x& = 1 + r - 2s, & \\y& = 3 - r + 2s, \\z& = -5 - 4r;\ r,\; s\in \mathbb{R} \\ \end{aligned} \]
\(\gamma \colon 3x - 2 = 0\)
\(\beta \colon 2z - 2 = 0\)
\(\alpha \colon x + y - 2 = 0\)

9000106603

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[ \begin{aligned}[t] p\colon x& = -1 - t, & \\y & = 11 - 2t, \\z & = 1 + t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}[t] q\colon x& = -3 + s, & \\y & = 4 - s, \\z & = 6 + 2s;\ s\in \mathbb{R} \\ \end{aligned} \]
intersecting lines
parallel lines, not identical
identical lines
skew lines

9000106604

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[ \begin{aligned}[t] p\colon x& = 1 + 3t& \\y & = 2 - 6t \\z & = 3t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}[t] q\colon x& = 4 - 2s & \\y & = 1 + 4s \\z & = 3 - 2s;\ s\in \mathbb{R} \\ \end{aligned} \]
parallel lines, not identical
identical lines
intersecting lines
skew lines

9000106605

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[ \begin{aligned}[t] p\colon x& = 5 - 3t, & \\y & = t, \\z & = 5 - t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}[t] q\colon x& = -4 + 3s,& \\y & = 3 - s, \\z & = 2 + s;\ s\in \mathbb{R} \\ \end{aligned} \]
identical lines
parallel lines, not identical
intersecting lines
skew lines

9000106606

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[ \begin{aligned}[t] p\colon x& = 2t, & \\y & = 3 - t, \\z & = 4 - t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}[t] q\colon x& = 2 - 2s, & \\y & = -1 + s, \\z & = 6 + 3s;\ s\in \mathbb{R} \\ \end{aligned} \]
skew lines
parallel lines, not identical
intersecting lines
identical lines

9000106607

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[\begin{aligned} p\colon &x = 2, &q\colon &x =\phantom{ -}1 -\phantom{ 3}s, & & & & \\ &y = 3 -\phantom{ 2}t, & &y =\phantom{ -}2 + 3s, & & & & \\ &z = 3 + 2t;\ t\in \mathbb{R}, & &z = -1 - 2s;\ s\in \mathbb{R} & & & & \end{aligned}\]
skew lines
parallel lines, not identical
intersecting lines
identical lines