Analytic geometry in a space

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 \(A = [0;0;1]\).
\(\varphi = 60^{\circ }\)
\(\varphi = 45^{\circ }\)
\(\varphi = 30^{\circ }\)
\(\varphi = 75^{\circ }\)

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

9000106603

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[ \begin{aligned}[t] p\colon x& = -1 - t, & \\y & = 11 - 2t, \\z & = 1 + t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}[t] q\colon x& = -3 + s, & \\y & = 4 - s, \\z & = 6 + 2s;\ s\in \mathbb{R} \\ \end{aligned} \]
intersecting lines
parallel lines, not identical
identical lines
skew lines

9000106604

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

9000106605

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[ \begin{aligned}[t] p\colon x& = 5 - 3t, & \\y & = t, \\z & = 5 - t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}[t] q\colon x& = -4 + 3s,& \\y & = 3 - s, \\z & = 2 + s;\ s\in \mathbb{R} \\ \end{aligned} \]
identical lines
parallel lines, not identical
intersecting lines
skew lines

9000106606

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