Space geometry

9000117401

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

9000117402

Level: 
A
Determine whether the following planes \(\rho \) and \(\sigma \) are parallel, identical or intersecting. \[ \begin{aligned}[t] \rho \colon &x = 2 + u - v, & \\&y = 1 + 2u + 4v, \\&z = -1 + 3u + 3v;\ u,v\in \mathbb{R}, \\ \end{aligned}\qquad \begin{aligned}[t] \sigma \colon &x = 2 + r - s, & \\&y = 7 + 2r + 4s, \\&z = 5 + 3r + 3s;\ s,t\in \mathbb{R}. \\ \end{aligned} \]
identical
parallel, not identical
intersecting

9000117404

Level: 
A
Determine whether the following planes are parallel, identical or intersecting. \[\begin{aligned} \rho \colon \frac{3} {8}x + \frac{1} {2}y -\frac{2} {3}z - 1 = 0,\qquad \sigma \colon \frac{3} {4}x + y -\frac{4} {3}z - 2 = 0 & & \end{aligned}\]
identical
parallel, not identical
intersecting

9000117406

Level: 
A
Determine whether the following planes are parallel, identical or intersecting. \[\begin{aligned} \rho \colon \frac{3} {2}x -\frac{1} {4}y + \frac{2} {3}z -\frac{2} {5} = 0,\qquad \sigma \colon \frac{2} {3}x - 4y + \frac{3} {2}z -\frac{5} {2} = 0 & & \end{aligned}\]
intersecting
identical
parallel, not identical

9000117408

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

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