Space geometry

1103189001

Level: 
B
Find the general form of the equation of the plane \( \alpha \) that is perpendicular to the straight line \( p \) given by: \begin{align*} x&=7+t, \\ y&=2t, \\ z&=4-t;\ t\in\mathbb{R}, \end{align*} and passes through the point \( A=[1;0;4] \). Consequently, find the coordinates of the point \( B \) which is the point of intersection of \( p \) and \( \alpha \) (see the picture).
\( \alpha\colon x+2y-z+3=0;\ B=[6;-2;5] \)
\( \alpha\colon x+2y-z-3;\ B=[6;-2;5] \)
\( \alpha\colon x+2y-z-3=0;\ B=[8;2;3] \)
\( \alpha\colon x+2y-z+3=0;\ B=[8;2;3] \)

1103189002

Level: 
B
Find the general form of the equation of the plane \( \beta \) that passes through the points \( M=[-1;1;-3] \) and \( N=[0;2;-1] \) and is perpendicular to the plane \( \alpha \) given by \( 3x-y+2=0 \) (see the picture).
\( \beta\colon x+3y-2z-8=0 \)
\( \beta\colon x+3z+10=0 \)
\( \beta\colon x+3z+3=0 \)
\( \beta\colon x+3y-2z+8=0 \)

1103189003

Level: 
B
Find the general form of the equation of the plane \( \beta \) that passes through the straight line \( p \) given by parametric equations \begin{align*} x&=1+2t, \\ y&=-2t, \\ z&=1+t;\ t\in\mathbb{R}, \end{align*} and is perpendicular to the plane \( \alpha \) given by \( x+3y-z-7=0 \) (see the picture).
\( \beta\colon x-3y-8z+7=0 \)
\( \beta\colon 2x-2y+z-3=0 \)
\( \beta\colon x-3y-8z-7=0 \)
\( \beta\colon 2x-2y+z+3=0 \)

1103189004

Level: 
B
We are given the point \( A=[2;-1;-4] \) and planes \( \rho \) by \( x-y+3z-5=0 \) and \( \sigma \) by \( 2x-y-z-8=0 \). Find the general form of the equation of the plane \( \alpha \) which passes through the point \( A \) and is perpendicular to both planes (see the picture).
\( \alpha\colon 4x+7y+z+3=0 \)
\( \alpha\colon -2x+5y-3z-3=0 \)
\( \alpha\colon 4x-7y+z+3=0 \)
\( \alpha\colon 2x-5y+3z+3=0 \)

2010005005

Level: 
B
Given points \(C = [-2;3;-1]\), \(D= [1;2;-3]\), find the angle between the line \(CD\) and the line \(p\). \[ \begin{aligned}p\colon x& = 2 -s, & \\y & = 3, \\z & = 2s;\ s\in \mathbb{R} \\ \end{aligned} \] Round your answer to the nearest minute.
\(33^{\circ }13'\)
\(56^{\circ }47'\)
\(90^{\circ }\)
\(146^{\circ }47'\)

2010008701

Level: 
B
We are given the points \(K = [ 1; −2; 1]\), \(L = [2; 0; −3]\) and the plane \(\rho\) by \(x-2z+3=0\). Find the general form of the equation of the plane \(\sigma\) in which the line \(KL\) is located and is perpendicular to the plane \(\rho\) (see the picture).
\( \sigma\colon 2x+y+z-1=0 \)
\( \sigma\colon 2x+3y+2z+2=0 \)
\( \sigma\colon 2y+z+3=0 \)
\( \sigma\colon 2x+y-4=0 \)

2010008702

Level: 
B
We are given the point \( P=[3;-4;-5] \) and planes \( \alpha \) by \( 2x-y-3z-5=0 \) and \( \beta \) by \( 3x-2y-4z+3=0 \). Find the general form of the equation of the plane \( \sigma \) which passes through the point \( P \) and is perpendicular to both planes \(\alpha\) and \(\beta\) (see the picture).
\( \sigma\colon 2x+y+z+3=0 \)
\( \sigma\colon 2x-y-z+15=0 \)
\( \sigma\colon 2x-y+z-5=0 \)
\( \sigma\colon 2x+y-z-7=0 \)