Analytical Space Geometry

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.
$70^{\circ}32'$
$54^{\circ}44'$
$45^{\circ}$
$51^{\circ}4'$

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.
$1$
$\sqrt3$
$\frac{\sqrt3}2$
$\frac{\sqrt5}2$

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 $ACH$.
$\frac{2\sqrt3}3$
$\frac{\sqrt3}3$
$\frac{2\sqrt6}3$
$\frac23$

1003189005

Level: 
B
We are given a straight line \( p \) by parametric equations \begin{align*} x&=1+t, \\ y&= 1+2t, \\ z&= 4-t,\ t\in\mathbb{R}. \end{align*} Find the parametric equations of the line \( p' \) that is an orthogonal projection of the line \( p \) into the coordinate \(xy\)-plane .
$\begin{aligned} p'\colon x&=5+s, \\ y&= 9+2s, \\ z&= 0,\ s\in\mathbb{R} \end{aligned}$
$\begin{aligned} p'\colon x&=5+s, \\ y&= 9-2s, \\ z&=0,\ s\in\mathbb{R} \end{aligned}$
$\begin{aligned} p'\colon x&=1+s, \\ y&=1+2s, \\ z&= 4,\ s\in\mathbb{R} \end{aligned}$
$\begin{aligned} p'\colon x&=5+2s, \\ y&=9+s, \\ z&= 0,\ s\in\mathbb{R} \end{aligned}$

1103189004

Level: 
B
We are given the point \( A=[2,-1,-4] \) and planes \( \rho \) by \( x-y+3z-5=0 \) and \( \sigma \) by \( 2x-y-z-8=0 \). Find the general form of the equation of the plane \( \alpha \) which passes through the point \( A \) and is perpendicular to both planes (see the picture).
\( \alpha\colon 4x+7y+z+3=0 \)
\( \alpha\colon -2x+5y-3z-3=0 \)
\( \alpha\colon 4x-7y+z+3=0 \)
\( \alpha\colon 2x-5y+3z+3=0 \)

1103189003

Level: 
B
Find the general form of the equation of the plane \( \beta \) that passes through the straight line \( p \) given by parametric equations \begin{align*} x&=1+2t, \\ y&=-2t, \\ z&=1+t,\ t\in\mathbb{R}, \end{align*} and is perpendicular to the plane \( \alpha \) given by \( x+3y-z-7=0 \) (see the picture).
\( \beta\colon x-3y-8z+7=0 \)
\( \beta\colon 2x-2y+z-3=0 \)
\( \beta\colon x-3y-8z-7=0 \)
\( \beta\colon 2x-2y+z+3=0 \)

1103189002

Level: 
B
Find the general form of the equation of the plane \( \beta \) that passes through the points \( M=[-1,1,-3] \) and \( N=[0,2,-1] \) and is perpendicular to the plane \( \alpha \) given by \( 3x-y+2=0 \) (see the picture).
\( \beta\colon x+3y-2z-8=0 \)
\( \beta\colon x+3z+10=0 \)
\( \beta\colon x+3z+3=0 \)
\( \beta\colon x+3y-2z+8=0 \)

1103189001

Level: 
B
Find the general form of the equation of the plane \( \alpha \) that is perpendicular to the straight line \( p \) given by: \begin{align*} x&=7+t, \\ y&=2t, \\ z&=4-t,\ t\in\mathbb{R}, \end{align*} and passes through the point \( A=[1,0,4] \). Consequently, find the coordinates of the point \( B \) which is the point of intersection of \( p \) and \( \alpha \) (see the picture).
\( \alpha\colon x+2y-z+3=0,\ B=[6,-2,5] \)
\( \alpha\colon x+2y-z-3,\ B=[6,-2,5] \)
\( \alpha\colon x+2y-z-3=0,\ B=[8,2,3] \)
\( \alpha\colon x+2y-z+3=0,\ B=[8,2,3] \)