Analytical 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 \(30^{\circ }\).
\(\begin{aligned}[t] p\colon x& = 2 + t, & \\y & = 1 + 3t, \\z & = -2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] r\colon x& = -2t, & \\y & = -3 + t, \\z & = 1 - 3t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 2 + 3t, & \\y & = 3 - 2t, \\z & = 3 + t,\ t\in \mathbb{R} \\ \end{aligned}\)

9000106601

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[ \begin{aligned}[t] p\colon x& = -6 - t,& \\y & = 7 + t, \\z & = -2t,\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}[t] q\colon x& = -1 - 2s, & \\y & = 2 + 2s, \\z & = 10 - 4s,\ s\in \mathbb{R} \\ \end{aligned} \]
identical lines
parallel lines, not identical
intersecting lines
skew lines

9000106602

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

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

9000106608

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

9000106301

Level: 
B
Find the line $k$ which is perpendicular to the plane \(\alpha \) \[ \alpha \colon 2x + y - z - 5 = 0 \] and passes through the point \(A = [0,0,1]\).
\(\begin{aligned}[t] x& =\phantom{ 1 -} 2t, & \\y& =\phantom{ 1 -}\ t, \\z& = 1 - t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2 + 2m, & \\y& =\phantom{ -}1 +\phantom{ 2}m, \\z& = -1 -\phantom{ 2}m,\ m\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2k, & \\y& =\phantom{ -2}k, \\z& = -\phantom{2}k,\ k\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2, & \\y& =\phantom{ -}1, \\z& = -1 + u,\ u\in \mathbb{R} \\ \end{aligned}\)