Analytical space geometry

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 \( \rho \).
\( \rho\colon x-y+z=0;\ |M\rho|=\sqrt3 \)
\( \rho\colon x-y+z+2=0;\ |M\rho|=\sqrt3 \)
\( \rho\colon x-y+z+2=0;\ |M\rho|=2\sqrt3 \)
\( \rho\colon x-y+z=0;\ |M\rho|=2\sqrt3 \)

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.
\( \begin{aligned} p'\colon x&=8+4t, \\ y&=3t, \\ z&=5+3t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=8+4t, \\ y&=0, \\ z&=5+3t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=8+4t, \\ y&=-3t, \\ z&=5+3t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p'\colon x&=8-4t, \\ y&=3t, \\ z&=5-3t;\ t\in\mathbb{R} \end{aligned} \)

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

1003124006

Level: 
A
Find the value of a parameter \( a\in\mathbb{R} \) so that the point \( D=[-2;1;1] \) lies on the straight line \( p \) defined by the parametric equations \[\begin{aligned}p\colon x&=1+m,\\ y&=-2+m,\\ z&=a+m;\ m\in\mathbb{R}. \end{aligned}\]
no such values of \(a \) exists
\( a=-1 \)
\( a=0 \)
\( a = 1\)

1003124005

Level: 
A
Find the value of a parameter \( a\in\mathbb{R} \) so that the point \( C=[2;0;6] \) lies on the straight line \( p \) defined by the parametric equations \[\begin{aligned}p\colon x&=-1+m,\\ y&=a+m,\\ z&=3+m;\ m\in\mathbb{R}.\end{aligned}\]
\( a=-3 \)
\( a=0 \)
\( a=-1 \)
no such values of \(a \) exists

1003124004

Level: 
A
Find the value of a parameter \( a\in\mathbb{R} \) so that the point \( B=[1;4;5] \) lies on the straight line \( p \) defined by the parametric equations \[\begin{aligned}p\colon x&=-1+m,\\ y&=2+am,\\ z&=3+m;\ m\in\mathbb{R}. \end{aligned}\]
\( a=1 \)
\( a=-1 \)
\( a=2 \)
no such value of \( a \) exists

1003124003

Level: 
A
Find the missing coordinates of the point \( B=[x_B; y_B;-3] \) lying on a straight line \( p \) defined by the parametric equations \[\begin{aligned} p\colon x&=-1+\frac14m,\\ y&=2+m,\\ z&=5-m;\ m\in\mathbb{R}.\end{aligned} \]
\( B=[1;10;-3] \)
\( B=[-3;-6;-3] \)
\( B=[1;3;-3] \)
\( B=[-3;6;-3] \)

1003124002

Level: 
A
From the given options choose the parametric equations which describe a straight line \( p \) passing through the points \( A=[-2;0;1] \) and \( B=[2;0;-3] \).
\( \begin{aligned} p\colon x&=2-t, \\ y&=0, \\ z&=-3+t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=2+4t, \\ y&=0, \\ z&=-3+4t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=2, \\ y&=0, \\ z&=-3+t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=2-2t, \\ y&=0, \\ z&=-3+t;\ t\in\mathbb{R} \end{aligned} \)

1003124001

Level: 
A
We are given a straight line \( q=\left\{[3t;2-2t;1+t]\text{, }t\in\mathbb{R}\right\} \) and four points \( A=[-6;6;-1] \), \( B=[-3;0;0] \), \( C=[0;2;1] \) and \( D=[3;0;2] \). Out of the given points select all that lie on the straight line \( q \). (Choose the corresponding option.)
\( A \), \( C \), \( D \)
\( B \), \( C \), \( D \)
\( B \), \( C \)
\( A \), \( B \), \( C \)