Analytic geometry in a space

1003188803

Level: 
A
A plane \( \rho \) is defined by the point \( A=[3;1;1] \) and a straight line \( p \) defined by the following parametric equations: \begin{align*} p\colon x&=4+4t, \\ y&=-1-2t, \\ z&=1+t;\ t\in\mathbb{R} \end{align*} Find the parametric equations of the plane \( \rho \).
$\begin{aligned} \rho\colon x&=4+4t+s, \\ y&=-1-2t-2s, \\ z&=1+t;\ t,s\in\mathbb{R} \end{aligned}$
$\begin{aligned} \rho\colon x&=4+4t+3s, \\ y&=-1-2t+s, \\ z&=1+t+s;\ t,s\in\mathbb{R} \end{aligned}$
$\begin{aligned} \rho\colon x&=3+4t+4s, \\ y&=1-2t-s, \\ z&=1+t+s;\ t,s\in\mathbb{R} \end{aligned}$
$\begin{aligned} \rho\colon x&=3+4t-4s, \\ y&=1-2t+2s, \\ z&=1+t-s;\ t,s\in\mathbb{R} \end{aligned}$

1003188802

Level: 
A
Find the missing coordinates of the points\( M=[2;m;0] \) and \( N=[0;3;n] \) so that they lie on a plane \( \rho \) defined by the following parametric equations: \begin{align*} \rho\colon x&=4+2s, \\ y&=-1-2t, \\ z&=1+t+s;\ t,s\in\mathbb{R} \end{align*} Choose the option in which values of both \( m \) and \( n \) are correct.
\( m=-1 \), \( n=-3 \)
\( m=-1 \), \( n=3 \)
\( m=1 \), \( n=-3 \)
\( m=1 \), \( n=3 \)

1003188801

Level: 
A
We are given points \( A=[2;4;0] \), \( B=[4;-1;1] \) and \( C=[0;1;1] \). From the following list, choose the parametric equations which represent a plane \( \rho \) defined by the points \( A \), \( B \), and \( C \).
$\begin{aligned} \rho\colon x&=4+2t+2s, \\ y&=-1-t-5s, \\ z&=1+s;\ t,s\in\mathbb{R} \end{aligned}$
$\begin{aligned} \rho\colon x&=4+4t+2s, \\ y&=-1-2t-5s, \\ z&=1+t+s;\ t,s\in\mathbb{R} \end{aligned}$
$\begin{aligned} \rho\colon x&=2t+4s, \\ y&=1-t-2s, \\ z&=1;\ t,s\in\mathbb{R} \end{aligned}$
$\begin{aligned} \rho\colon x&=2t-2s, \\ y&=1-5t+5s, \\ z&=1+t-s;\ t,s\in\mathbb{R} \end{aligned}$

1103188706

Level: 
A
We are given points \( A=[2;4;0] \) and \( B=[4;7;6] \). Find parametric equations of a line \( q \), which is the orthogonal projection of the line \( AB \) into the coordinate plane \( xy \).
$\begin{aligned} p\colon x&=4+2t, \\ y&=7+3t, \\ z&=0;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=2+4t, \\ y&=4+7t, \\ z&=6t;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=4+2t, \\ y&=7+3t, \\ z&=6;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=2-2t, \\ y&=4-3t, \\ z&=-6t;\ t\in\mathbb{R} \end{aligned}$

1103188705

Level: 
A
Find parametric equations of the line \( p \) that passes through the point \( K=[4;2;3] \), is parallel to the \( xy \)-coordinate plane, and is intersecting the \( z \)-axis.
$\begin{aligned} p\colon x&=4+2t, \\ y&=2+t, \\ z&=3;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=4+2t, \\ y&=2+t, \\ z&=3+t;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=4, \\ y&=2, \\ z&=3+3t;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=4-2t, \\ y&=2-4t, \\ z&=3t;\ t\in\mathbb{R} \end{aligned}$

1003188704

Level: 
A
Given points \( A=[-4;1;4] \) and \( B=[4;-3;0] \), determine which of the following parametric equations does not define the ray \( AB \).
$\begin{aligned} \mapsto AB\colon x&=-4+8t, \\ y&=1-4t, \\ z&=4-4t;\ t\in(-\infty;0] \end{aligned}$
$\begin{aligned} \mapsto AB\colon x&=-4+8t, \\ y&=1-4t, \\ z&=4-4t;\ t\in[0;\infty) \end{aligned}$
$\begin{aligned} \mapsto AB\colon x&=-4+2t, \\ y&=1-t, \\ z&=4-t;\ t\in[0;\infty) \end{aligned}$
$\begin{aligned} \mapsto AB\colon x&=-4-8t, \\ y&=1+4t, \\ z&=4+4t;\ t\in(-\infty;0] \end{aligned}$

1003188703

Level: 
A
Given points \( A=[-4;1;4] \) and \( B=[4;-3;0] \), determine which of the following parametric equations do not define the line segment \( AB \).
$\begin{aligned} AB\colon x&=-4+8t, \\ y&=1+4t, \\ z&=4-4t,\ t\in[0;1] \end{aligned}$
$\begin{aligned} AB\colon x&=-4+8t, \\ y&=1-4t, \\ z&=4-4t,\ t\in[0;1] \end{aligned}$
$\begin{aligned} AB\colon x&=4+8t, \\ y&=-3-4t, \\ z&=-4t,\ t\in[-1;0] \end{aligned}$
$\begin{aligned} AB\colon x&=-4+2t, \\ y&=1-t, \\ z&=4-t,\ t\in[0;4] \end{aligned}$

1003188702

Level: 
A
We are given points \( A=[-2;3;0] \), \( B=[6;1;6] \) and \( C=[1;0;4] \). Find the parametric equations of a line \( p \) that passes through the point \( C \) and through the midpoint of the line segment \( AB \).
$\begin{aligned} p\colon x&=1+t, \\ y&=2t, \\ z&=4-t;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=1+2t, \\ y&=-t, \\ z&=4-t;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=1-t, \\ y&=2t, \\ z&=4+t;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=1+2t, \\ y&=t, \\ z&=4+t;\ t\in\mathbb{R} \end{aligned}$