Analytical Space Geometry

9000106610

Level: 
A
Determine whether two lines are identical, parallel, intersecting or skew. The first line is the line passes through the points \(A = [1,-4,2]\) and \(B = [3,0,0]\) and the second line is the line passes through the points \(C = [3,-5,5]\) and \(D = [-1,-3,-1]\).
intersecting lines
parallel lines, not identical
identical lines
skew lines

9000106304

Level: 
B
Find the third coordinate of the point \(B = [2,0,?]\) using the fact that this point is in the plane \(\alpha \) defined by the equation \[ \alpha \colon 2x + y - z - 5 = 0. \] Use the point \(B\) to find the angle \(\varphi \) between the plane \(\alpha \) and the line \(AB\), where \(A = [0,0,1]\).
\(\varphi = 60^{\circ }\)
\(\varphi = 45^{\circ }\)
\(\varphi = 30^{\circ }\)
\(\varphi = 75^{\circ }\)

9000106305

Level: 
B
Find the area of the triangle \(ABS\). Only first two coordinates of the point $B=[2,0,?]$ are given and $B$ lies in the plane $\alpha$ defined by the equation \[ \alpha \colon 2x + y - z - 5 = 0. \] The point \(S\) is the intersection point of the plane \(\alpha \) and the line \(k\) which is perpendicular to \(\alpha \) and passes through the point \(A = [0,0,1]\).
\(\sqrt{3}\)
\(2\)
\(4\)
\(\sqrt{6}\)

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 \(\alpha \). \[ \alpha \colon 2x + y - z - 5 = 0 \]
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 11& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 10& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 12& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z + 1& = 0& \\2x + y - z - 9& = 0 \\ \end{aligned}\)

9000106601

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[ \begin{aligned}[t] p\colon x& = -6 - t,& \\y & = 7 + t, \\z & = -2t,\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}[t] q\colon x& = -1 - 2s, & \\y & = 2 + 2s, \\z & = 10 - 4s,\ s\in \mathbb{R} \\ \end{aligned} \]
identical lines
parallel lines, not identical
intersecting lines
skew lines

9000106602

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[ \begin{aligned}[t] p\colon x& = -3 + 2t,& \\y & = 1 - t, \\z & = 3 - 2t,\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}[t] q\colon x& = 2 - 4s, & \\y & = -3 + 2s, \\z & = 6 + 4s,\ s\in \mathbb{R} \\ \end{aligned} \]
parallel lines, not identical
identical lines
intersecting lines
skew lines

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