Space geometry

1103212201

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

1103212202

Level: 
C
A straight line \( p \) is given by the points \( M=[4;3;2] \) and \( N=[0;6;7] \) (see the picture). Find the parametric equations of the line \( p' \) that is symmetrical to the line \( p \) in the plane symmetry across the coordinate \( yz \)-plane.
\( \begin{aligned} p'\colon x&=4t, \\ y&=6+3t, \\ z&=7+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=-4t, \\ y&=6+3t, \\ z&=7+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=4t, \\ y&=6-3t, \\ z&=7+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=-4t, \\ y&=6-3t, \\ z&=7+5t;\ t\in\mathbb{R} \end{aligned} \)

1103212203

Level: 
C
A straight line \( p \) is given by the points \( M=[4;3;2] \) and \( N=[8;0;5] \) (see the picture). Find the parametric equations of the line \( p' \) that is symmetrical to the line \( p \) in the plane symmetry across the coordinate \( xz \)-plane.
\( \begin{aligned} p'\colon x&=8+4t, \\ y&=3t, \\ z&=5+3t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=8+4t, \\ y&=0, \\ z&=5+3t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=8+4t, \\ y&=-3t, \\ z&=5+3t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=8-4t, \\ y&=3t, \\ z&=5-3t;\ t\in\mathbb{R} \end{aligned} \)

1103212204

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) is placed in a coordinate system (see the picture). Let the point \( M \) be the centre of the edge \( EF \). Find the general form equation of the plane \( \rho \) passing through the points \( B \), \( D \), and \( G \) and calculate the distance of \( M \) from the plane \( \rho \).
\( \rho\colon x-y+z=0;\ |M\rho|=\sqrt3 \)
\( \rho\colon x-y+z+2=0;\ |M\rho|=\sqrt3 \)
\( \rho\colon x-y+z+2=0;\ |M\rho|=2\sqrt3 \)
\( \rho\colon x-y+z=0;\ |M\rho|=2\sqrt3 \)

1103212205

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

1103212206

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) is placed in a coordinate system (see the picture). Let \( p \) be a line of intersection of planes \( \alpha \) and \( \beta \), where \( \alpha \) is passing through \( C \), \( F \) and \( H \) and \( \beta \) is passing through \( A \), \( F \) and \( H \). Find the parametric equations of the line \( p \) and calculate the angle \( \varphi \) between planes \( \alpha \) and \( \beta \) . Round \( \varphi \) to the nearest minute.
\( \begin{aligned} p\colon x&=t, & \varphi&\doteq 70^{\circ}32' \\ y&=t, & &\\ z&=2;\ t\in\mathbb{R}, & & \end{aligned} \)
\( \begin{aligned} p\colon x&=2t, & \varphi&\doteq 90^{\circ} \\ y&=2t, & & \\ z&=2+2t;\ t\in\mathbb{R}, & & \end{aligned} \)
\( \begin{aligned} p\colon x&=t, & \varphi&\doteq 90^{\circ}\\ y&=t, & & \\ z&=2;\ t\in\mathbb{R}, & & \end{aligned} \)
\( \begin{aligned} p\colon x&=2t, & \varphi&\doteq 70^{\circ}32' \\ y&=2t, & & \\ z&=2t;\ t\in\mathbb{R}, & & \end{aligned} \)

1103212901

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) units is placed in a coordinate system (see the picture). Find the distance of parallel lines \( p=KL \) and \( q=MN \), where points \( K \), \( L \), \( M \) and \( N \) are midpoints of edges \( CD \), \( BC \), \( EH \) and \( EF \) respectively.
\( |pq|=\sqrt6 \)
\( |pq|=2\sqrt3 \)
\( |pq|=3\sqrt2 \)
\( |pq|=2\sqrt2 \)

1103212902

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) units is placed in a coordinate system (see the picture). Let \( S \) be the midpoint of the face \( ABFE \), and let \( K \) and \( L \) be the midpoints of edges \( DH \) and \( CG \) consecutively. Find the standard equation of a plane \( \alpha \) passing through the points \( A \), \( B \) and \( L \), and calculate the distance of the point \( S \) from \( \alpha \).
\( \alpha\colon x+2z-2=0;\ |S\alpha|=\frac{2\sqrt5}{5} \)
\( \alpha\colon x+2z-2=0;\ |S\alpha|=\frac{2\sqrt3}{3} \)
\( \alpha\colon x+2y-2=0;\ |S\alpha|=\frac{2\sqrt5}{5} \)
\( \alpha\colon x+2y-2=0;\ |S\alpha|=\frac{2\sqrt3}{3} \)

1103212903

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) units is placed in a coordinate system (see the picture). Find an angle \( \varphi \) between the plane \( \alpha \) passing through the points \( E \), \( D \) and \( C \) and the straight line \( AF \). Hint: An angle between a line and a plane is an angle between the line and its orthogonal projection into this plane.
\( \varphi = 30^{\circ} \)
\( \varphi = 15^{\circ} \)
\( \varphi = 45^{\circ} \)
\( \varphi = 60^{\circ} \)

1103212904

Level: 
C
A rectangle-based right pyramid \( ABCDV \) with a bottom edge length of \( 6 \) units and the perpendicular height of \( 6 \) units is placed in a coordinate system (see the picture). Let \( S \) be the midpoint of the edge \( AD \). Find the standard equation of the plane \( \alpha \) passing through the points \( B \), \( V \) and \( C \), and calculate the distance of the point \( S \) from \( \alpha \).
\( \alpha\colon 2y+z-12=0;\ d=|S\alpha|=\frac{12\sqrt5}{5} \)
\( \alpha\colon 2x+z-12=0;\ d=|S\alpha|=\frac{12\sqrt5}{5} \)
\( \alpha\colon 2y+z-12=0;\ d=|S\alpha|=\frac{6\sqrt5}{5} \)
\( \alpha\colon 2x+z-12=0;\ d=|S\alpha|=\frac{6\sqrt5}{5} \)