Analytic geometry in a space

1003233607

Level: 
C
Determine the relative position of three planes: \begin{align*} \alpha\colon\ &2x+y+9z-18=0, \\ \beta\colon\ &x+3y+2z+16=0, \\ \gamma\colon\ &x+2y+3z+6=0. \end{align*}
Planes $\alpha$, $\beta$ and $\gamma$ intersect in a straight line.
Each of the two planes are intersecting and the lines of intersection are three different lines parallel to each other.
All three planes intersect at just one point.

1003233605

Level: 
C
We are given skew lines $p$ and $q$. \begin{align*} p\colon x&= 1-t, & q\colon x&= 1-2s, \\ y&= 1+t, & y&=s, \\ z&= 3+2t;\ t\in\mathbb{R}, & z&= 3+3s;\ s\in\mathbb{R}. \end{align*} Find parametric equations of a straight line $r$, that is intersecting both lines $p$ and $q$ and lying in the plane $x+2y-z+2=0$.
$\begin{aligned} r\colon x&=-1+2m, \\ y&=3-3m, \\ z&=7-4m;\ m\in\mathbb{R} \end{aligned}$
$\begin{aligned} r\colon x&=-1+m, \\ y&=3+3m, \\ z&=7-m;\ m\in\mathbb{R} \end{aligned}$
$\begin{aligned} r\colon x&=-1+3m, \\ y&=3+2m, \\ z&=7+5m;\ m\in\mathbb{R} \end{aligned}$
$\begin{aligned} r\colon x&=-1+m, \\ y&=3-m, \\ z&=7+m;\ m\in\mathbb{R} \end{aligned}$

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 .
$\begin{aligned} p'\colon x&=5+s, \\ y&= 9+2s, \\ z&= 0;\ s\in\mathbb{R} \end{aligned}$
$\begin{aligned} p'\colon x&=5+s, \\ y&= 9-2s, \\ z&=0;\ s\in\mathbb{R} \end{aligned}$
$\begin{aligned} p'\colon x&=1+s, \\ y&=1+2s, \\ z&= 4;\ s\in\mathbb{R} \end{aligned}$
$\begin{aligned} p'\colon x&=5+2s, \\ y&=9+s, \\ z&= 0;\ s\in\mathbb{R} \end{aligned}$