Space geometry

9000106306

Level: 
B
Find the general equation of the plane which is perpendicular to the plane \(\alpha \) \[ \alpha \colon 2x + y - z - 5 = 0 \] and contains the line \(AB\), where \(A = [0;0;1]\) and \(B\) is a point in \(\alpha \) defined by it's first two coordinates \[ B = [2;0;?]. \]
\(x - y + z - 1 = 0\)
\(x + y - z + 1 = 0\)
\(2x - y + z - 1 = 0\)
\(- 2x + y - z + 1 = 0\)

9000106307

Level: 
C
Given points \(A = [0;0;1]\), \(B = [2;0;-1]\) and \(S = [2;1;0]\), find the parametric equations of the image of the line \(AB\) in a point reflection about the point \(S\).
\(\begin{aligned}[t] x& =\phantom{ -}4 + t, & \\y& =\phantom{ -}2, \\z& = -1 - t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& = 2 + 2m, & \\y& = 2 +\phantom{ 2}m, \\z& = 1 -\phantom{ 2}m;\ m\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}4 + 2k, & \\y& =\phantom{ -}2 +\phantom{ 2}k, \\z& = -1 -\phantom{ 2}k;\ k\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& = -2 + 2u, & \\y& =\phantom{ -}2, \\z& =\phantom{ -}1 - 2u;\ u\in \mathbb{R} \\ \end{aligned}\)

9000106308

Level: 
B
In the following list identify a pair of planes such that the distance of planes from the plane $\alpha$ is the same as the distance between the point $A=[0;0;1]$ and the plane \(\alpha \). \[ \alpha \colon 2x + y - z - 5 = 0 \]
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 11& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 10& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 12& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z + 1& = 0& \\2x + y - z - 9& = 0 \\ \end{aligned}\)

9000101904

Level: 
B
Find the angle between the \(x\)-axis and the line \(p\). \[ \begin{aligned}p\colon x& = 2 - t, & \\y & = 3t, \\z & = 1;\ t\in \mathbb{R} \\ \end{aligned} \] Round your answer to the nearest minute.
\(71^{\circ }34'\)
\(0^{\circ }\)
\(69^{\circ }17'\)
\(90^{\circ }\)

9000101909

Level: 
B
Given points \(A = [1;0;2]\), \(B = [1;0;0]\) and the plane \(\alpha \), \[ \alpha \colon 2x - 4y = 0, \] find the angle between the line \(AB\) and the plane \(\alpha \). Round your answer to the nearest minute.
\(0^{\circ }\)
\(22^{\circ }48'\)
\(45^{\circ }19'\)
\(90^{\circ }\)

9000101910

Level: 
B
The points \(A = [0;5;0]\), \(B = [5;5;0]\), \(C = [5;0;0]\) and \(D = [0;0;0]\) define the cube \(ABCDEFGH\). Find the angle between the line \(BF\) and the plane \(AFE\). Round your answer to the nearest minute.
\(0^{\circ }\)
\(35^{\circ }16'\)
\(45^{\circ }\)
\(90^{\circ }\)

9000101907

Level: 
B
The general plane \(\alpha \) has the equation \[ \alpha \colon 3z - 4 = 0 \] and the plane \(\beta \) has a normal vector \(\vec{n} = (0;0;1)\). Find the angle between \(\alpha \) and \(\beta \) and round your answer to the nearest degree.
\(0^{\circ }\)
\(30^{\circ }\)
\(45^{\circ }\)
\(90^{\circ }\)