Analytical space geometry

1103212905

Level: 
C
A rectangle-based right pyramid \( ABCDV \) with its bottom edge length of \( 6 \) units and the perpendicular height of \( 6 \) units is placed in a coordinate system (see the picture). Find the parametric equations of an intersection line \( p \) of planes \( \alpha \) and \( \beta \), where \( \alpha \) passes through the points \( B \), \( C \) and \( V \), and \( \beta \) passes through the points \( A \), \( D \) and \( V \). What is the measure of an angle \( \varphi \) between the planes \( \alpha \) and \( \beta \). Round \( \varphi \) to the nearest minute.
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 53^{\circ}8'\\ y&=3, &\\ z&=6;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 63^{\circ}8'\\ y&=3, &\\ z&=0;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 53^{\circ}8'\\ y&=3+t, &\\ z&=6+2t;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 63^{\circ}8'\\ y&=3, &\\ z&=6;\ t\in\mathbb{R} & \end{aligned}\)

1103233601

Level: 
C
Let $ABCDEFGH$ be a cube with an edge length of $1$ placed in the rectangular coordinate system. In the cube a regular tetrahedron $ACHF$ is highlighted (see the picture). Find its perpendicular height. \[ \] Hint: Find e.g. the distance between the point $F$ and the plane $ACH$.
$\frac{2\sqrt3}3$
$\frac{\sqrt3}3$
$\frac{2\sqrt6}3$
$\frac23$

1103233602

Level: 
C
Let $ABCDEFGH$ be a cube with an edge length of $1$ placed in the rectangular coordinate system. In the cube a regular tetrahedron $ACHF$ is highlighted (see the picture). Find the distance between the opposite edges of this tetrahedron.\[ \] Hint: A tetrahedron’s opposite edges lie on skew lines. Their distance is the same as the distance of the midpoint of one edge from the opposite edge.
$1$
$\sqrt3$
$\frac{\sqrt3}2$
$\frac{\sqrt5}2$

1103233603

Level: 
C
Let $ABCDEFGH$ be a cube with an edge length of $1$ placed in the rectangular coordinate system. In the cube a regular tetrahedron $ACHF$ is highlighted (see the picture). Find the angle between its faces and round the number to the nearest minute.
$70^{\circ}32'$
$54^{\circ}44'$
$45^{\circ}$
$51^{\circ}4'$

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.
\( \begin{aligned} q'\colon x&=4+2t, \\ y&=-6t, \\ z&=2+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+6t, \\ y&=6t, \\ z&=2+7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+2t, \\ y&=6t, \\ z&=2+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+6t, \\ y&=-6t, \\ z&=2+7t;\ t\in\mathbb{R} \end{aligned} \)

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 \( F \).
\( |\rho\sigma|=\sqrt3 \)
\( |\rho\sigma|=\frac{2\sqrt3}3 \)
\( |\rho\sigma|=\frac{3\sqrt3}2 \)
\( |\rho\sigma|=\frac{4\sqrt3}3 \)

2010008705

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 4 \) units is placed in a coordinate system (see the picture). Find the distance of parallel lines \( p=PQ\) and \( r=RS \), where points \( P \), \( Q \), \( R\) and \( S \) are midpoints of edges \(BF\), \(BC\), \(EH\) and \(DH\) respectively.
\( |pr|=2\sqrt6 \)
\( |pr|=4\sqrt3 \)
\( |pr|=6\sqrt2 \)
\( |pr|=4\sqrt2 \)

2010008706

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 4 \) units is placed in a coordinate system (see the picture). Find an angle \( \psi \) between the plane \( \rho \) passing through the points \( B \), \( D \) and \( H \) and the straight line \( CF \). Hint: An angle between a line and a plane is an angle between the line and its orthogonal projection into this plane.
\( \psi = \frac{\pi}6 \)
\( \psi = \frac{\pi}{12} \)
\( \psi = \frac{\pi}4 \)
\( \psi = \frac{\pi}3 \)