Space geometry

9000101903

Level: 
B
Given points \(A = [-1;0;3]\), \(B = [0;2;0]\), find the angle between the line \(AB\) and the line \(m\). \[ \begin{aligned}m\colon x& = 1 + 2t, & \\y & = -3t, \\z & = 1;\ t\in \mathbb{R} \\ \end{aligned} \] Round your answer to the nearest minute.
\(72^{\circ }45'\)
\(0^{\circ }\)
\(48^{\circ }15'\)
\(90^{\circ }\)

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.
\(71^{\circ }34'\)
\(0^{\circ }\)
\(69^{\circ }17'\)
\(90^{\circ }\)

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.
\(0^{\circ }\)
\(30^{\circ }\)
\(45^{\circ }\)
\(90^{\circ }\)

9000101908

Level: 
B
Find the angle between the line \(p\) and the plane \(\alpha \). \[ \alpha \colon x-3z+5 = 0;\qquad \qquad \begin{aligned}[t] p\colon x& = 3, & \\y & = 3t, \\z & = 1 - t;\ t\in \mathbb{R} \\ \end{aligned} \] Round your answer to the nearest minute.
\(17^{\circ }27'\)
\(0^{\circ }\)
\(47^{\circ }33'\)
\(90^{\circ }\)

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.
\(0^{\circ }\)
\(35^{\circ }16'\)
\(45^{\circ }\)
\(90^{\circ }\)

9000106301

Level: 
B
Find the line $k$ which is perpendicular to the plane \(\alpha \) \[ \alpha \colon 2x + y - z - 5 = 0 \] and passes through the point \(A = [0;0;1]\).
\(\begin{aligned}[t] x& =\phantom{ 1 -} 2t, & \\y& =\phantom{ 1 -}\ t, \\z& = 1 - t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2 + 2m, & \\y& =\phantom{ -}1 +\phantom{ 2}m, \\z& = -1 -\phantom{ 2}m;\ m\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2k, & \\y& =\phantom{ -2}k, \\z& = -\phantom{2}k;\ k\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2, & \\y& =\phantom{ -}1, \\z& = -1 + u;\ u\in \mathbb{R} \\ \end{aligned}\)

9000106302

Level: 
B
The plane \(\alpha \) has equation \[ \alpha : 2x + y - z - 5 = 0. \] The line \(k\) passes through the point \(A = [0;0;1]\) and is perpendicular to \(\alpha \). Find the intersection \(S\) of the line \(k\) and the plane \(\alpha \).
\(S = [2;1;0]\)
\(S = [2;0;1]\)
\(S = [-2;1;0]\)
\(S = [-2;0;1]\)