Analytical Space Geometry

9000106608

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

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 \(A = [0,0,1]\).
\(\begin{aligned}[t] x& =\phantom{ 1 -} 2t, & \\y& =\phantom{ 1 -}\ t, \\z& = 1 - t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2 + 2m, & \\y& =\phantom{ -}1 +\phantom{ 2}m, \\z& = -1 -\phantom{ 2}m,\ m\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2k, & \\y& =\phantom{ -2}k, \\z& = -\phantom{2}k,\ k\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2, & \\y& =\phantom{ -}1, \\z& = -1 + u,\ u\in \mathbb{R} \\ \end{aligned}\)

9000106609

Level: 
A
Determine whether two lines are identical, parallel, intersecting or skew. The first line is the line passes through the points \(A = [3,-2,1]\) and \(B = [0,7,7]\) and the second line is the line passes through the points \(C = [5,-8,-3]\) and \(D = [6,-11,-5]\).
identical lines
parallel lines, not identical
intersecting lines
skew lines

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 \(\alpha \).
\(S = [2,1,0]\)
\(S = [2,0,1]\)
\(S = [-2,1,0]\)
\(S = [-2,0,1]\)

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 \[ B = [2,0,?]. \]
\(x - y + z - 1 = 0\)
\(x + y - z + 1 = 0\)
\(2x - y + z - 1 = 0\)
\(- 2x + y - z + 1 = 0\)