Analytic geometry in a space

2010008704

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 3 \) is placed in a coordinate system (see the picture). Find the distance between parallel planes \( \rho \) and \( \sigma \), where \( \rho \) is passing through \( D \), \( E \) and \( G \) and \( \sigma \) is passing through \( A \), \( C \) and \( F \).
\( |\rho\sigma|=\sqrt3 \)
\( |\rho\sigma|=\frac{2\sqrt3}3 \)
\( |\rho\sigma|=\frac{3\sqrt3}2 \)
\( |\rho\sigma|=\frac{4\sqrt3}3 \)

2010008703

Level: 
C
A straight line \( q \) is given by the points \( K=[6;6;7] \) and \( L=[4;0;2] \) (see the picture). Find the parametric equations of the line \( q' \) that is symmetrical to the line \( q \) in the plane symmetry across the coordinate \( xz \)-plane.
\( \begin{aligned} q'\colon x&=4+2t, \\ y&=-6t, \\ z&=2+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+6t, \\ y&=6t, \\ z&=2+7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+2t, \\ y&=6t, \\ z&=2+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+6t, \\ y&=-6t, \\ z&=2+7t;\ t\in\mathbb{R} \end{aligned} \)

2010008702

Level: 
B
We are given the point \( P=[3;-4;-5] \) and planes \( \alpha \) by \( 2x-y-3z-5=0 \) and \( \beta \) by \( 3x-2y-4z+3=0 \). Find the general form of the equation of the plane \( \sigma \) which passes through the point \( P \) and is perpendicular to both planes \(\alpha\) and \(\beta\) (see the picture).
\( \sigma\colon 2x+y+z+3=0 \)
\( \sigma\colon 2x-y-z+15=0 \)
\( \sigma\colon 2x-y+z-5=0 \)
\( \sigma\colon 2x+y-z-7=0 \)

2010008701

Level: 
B
We are given the points \(K = [ 1; −2; 1]\), \(L = [2; 0; −3]\) and the plane \(\rho\) by \(x-2z+3=0\). Find the general form of the equation of the plane \(\sigma\) in which the line \(KL\) is located and is perpendicular to the plane \(\rho\) (see the picture).
\( \sigma\colon 2x+y+z-1=0 \)
\( \sigma\colon 2x+3y+2z+2=0 \)
\( \sigma\colon 2y+z+3=0 \)
\( \sigma\colon 2x+y-4=0 \)

2010008908

Level: 
C
We are given skew lines $a$ and $b$. \begin{align*} a\colon x&= -1-2t, & b\colon x&= 1-3s, \\ y&= -2+3t, & y&=2s, \\ z&= -4+2t;\ t\in\mathbb{R}, & z&= 2-2s;\ s\in\mathbb{R}. \end{align*} Find parametric equations of a straight line $p$, that is intersecting both lines $a$ and $b$ and lying in the plane $2x+3y-z-8=0$.
$\begin{aligned} p\colon x&=-9+r, \\ y&=10+r, \\ z&=4+5r;\ r\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=-9-2r, \\ y&=10-2r, \\ z&=4+10r;\ r\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=-9-10r, \\ y&=10+9r, \\ z&=4-r;\ r\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=-9+2r, \\ y&=10+2r, \\ z&=4-2r;\ r\in\mathbb{R} \end{aligned}$

2010008906

Level: 
A
We are given two intersecting planes \(2x - 3y + 5z - 9 = 0\) and \(3x - y + 2z - 1 = 0\). Find the parametric equations of their line of intersection \(p\).
\( \begin{aligned} p\colon x&=-1-t, \\ y&=-2+ 11t, \\ z&=1+ 7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=-1-11t, \\ y&=-2+ 11t, \\ z&=1+ 7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=-1+t, \\ y&=-2+ 11t, \\ z&=1- 11t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=-1-11t, \\ y&=-2+ 11t, \\ z&=1- 11t;\ t\in\mathbb{R} \end{aligned} \)

2010008905

Level: 
A
Determine the relative position of the plane \( \sigma \) with general equation \( x-2y+3z-1=0 \) and the straight line \( p \) with parametric equations: \[ \begin{aligned} x&=4, \\ y&=5+3t, \\ z&=2+2t;\ t\in\mathbb{R}. \end{aligned} \]
\( p\parallel\sigma,\ p\not{\!\!\subset} \sigma \)
\( p \subset \sigma \)
\( p \) is intersecting the plane \( \sigma \)

2010008904

Level: 
A
We are given points \( K=[4;0;3] \), \( L=[1;-3;2] \) and \( M=[2;2;0] \). From the following list, choose the parametric equations which represent a plane \( \sigma \) defined by the points \( K \), \( L \), and \( M \).
$\begin{aligned} \sigma\colon x&=1+3r+s, \\ y&=-3+3r+5s, \\ z&=2+r-2s;\ r,s\in\mathbb{R} \end{aligned}$
$\begin{aligned} \sigma\colon x&=1-3r-s, \\ y&=-3+3r-5s, \\ z&=2+r+2s;\ r,s\in\mathbb{R} \end{aligned}$
$\begin{aligned} \sigma\colon x&=1-3r+s, \\ y&=-3-3r+5s, \\ z&=2+r-2s;\ r,s\in\mathbb{R} \end{aligned}$
$\begin{aligned} \sigma\colon x&=1+3r+s, \\ y&=-3+3r-5s, \\ z&=2-r+2s;\ r,s\in\mathbb{R} \end{aligned}$