Analytical Space Geometry

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 \( AB \).
$\begin{aligned} p\colon x&=1+t, \\ y&=2t, \\ z&=4-t,\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=1+2t, \\ y&=-t, \\ z&=4-t,\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=1-t, \\ y&=2t, \\ z&=4+t,\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=1+2t, \\ y&=t, \\ z&=4+t,\ t\in\mathbb{R} \end{aligned}$

1003164406

Level: 
A
Determine whether any of the lines \( p \), \( q \) or \( r \) defined by the parametric equations given below passes through the coordinate origin. \begin{align*} p\colon x&=-2+4t, & q\colon x&=-5-5s, & r\colon x&=3-6u, \\ y&=1-2t, & y&=2-2s, & y&=-\frac12+u, \\ z&=-3+3t,\ t\in\mathbb R & z&=5+5s,\ s\in \mathbb R & z&=2-4u,\ u\in \mathbb R \end{align*}
Yes, it's the straight line \( r \).
Yes, it's the straight line \( p \).
Yes, it's the straight line \( q \).
None of the lines passes through the coordinate origin.

1003164405

Level: 
A
Determine whether the line \( p \) defined by parametric equations: \begin{align*} x&=-2+2t, \\ y&=1+3t, \\ z&=-3+3t,\ t\in\mathbb{R} \end{align*} intersects any of the coordinate axis.
Yes, it intersects the \( y \)-axis.
Yes, it intersects the \( x \)-axis.
Yes, it intersects the \( z \)-axis.
It intersects no coordinate axis.

1003164404

Level: 
A
Let a straight line \( p \) be defined by parametric equations: \begin{align*} x&=3+t, \\ y&=2-t, \\ z&=4,\ t\in\mathbb{R}. \end{align*} Find the coordinates of the intersection point \( M \) of the line \( p \) with the \( xy \)-coordinate plane.
There is no such point \( M \).
\( M=[0,0,4] \)
\( M=[-3,2,0] \)
\( M=[1,-1,0] \)