Analytical Space Geometry

9000117408

Level: 
B
In the following list find the plane perpendicular to the plane \(\rho \). \[\begin{aligned} \rho \colon 2x - 3y + 7z - 2 = 0 & & \end{aligned}\]
\(\omega \colon x + 3y + z + 7 = 0\)
\(\tau \colon - 2x + 3y - 7z + 2 = 0\)
\(\nu \colon - 2x - 3y + 7z + 2 = 0\)
\(\sigma \colon 7x - 3y + 2z - 2 = 0\)

9000117401

Level: 
B
Find the intersection of the planes \(\rho \) and \(\sigma \). \[\begin{aligned} \rho \colon 2x - 5y + 4z - 10 = 0,\qquad \sigma \colon x - y - z - 2 = 0 & & \end{aligned}\]
\(\begin{aligned}[t] p\colon x& = 3t, & \\y & = -2 + 2t, \\z & = t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 2s - 10,& \\y & = 5s - 10, \\z & = s,\ s\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] a\colon x& = 2u - 4,& \\y & = 2u - 4, \\z & = u,\ u\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] b\colon x& = 3v + 1,& \\y & = v - 2, \\z & = v,\ v\in \mathbb{R} \\ \end{aligned}\)

9000111802

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

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 \(\sqrt{5}\). \[ \begin{aligned}[t] s\colon x& = -1 + t,& \\y & = 2t, \\z & = 2 - t,\ t\in \mathbb{R} \\ \end{aligned} \]
\(\begin{aligned}[t] r\colon x& = 3 - 2t,& \\y & = 3 - 4t, \\z & = 2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 1, & \\y & = -1 + 5t, \\z & = 2 - 2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = -5 - t,& \\y & = 2 - 2t, \\z & = 2 + t,\ t\in \mathbb{R} \\ \end{aligned}\)

9000111806

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

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

9000111808

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