Project ID:
7300000024
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Question:
Spárujte každú dvojicu analytických vyjadrení priamok/rovín s informáciou o ich vzájomnej polohe.
Questions Title:
Analytické vyjadrenia priamok / rovín
Answers Title:
Vzájomná poloha priamok / rovín
Question 1:
$$\left.\begin{aligned}
x &= 3 - t \cr
y &= -2 - 2t\cr
z &= 1 + t\cr
\end{aligned}\ \right\}\ t \in\mathbb{R}
\qquad\qquad\left.\begin{aligned}
x &= -4 + s\cr
y &= -7+3s\cr
z &= -1-2s
\end{aligned}\ \right\}\ s\in\mathbb{R}
$$
Answer 1:
Priamky sú rôznobežné.
Question 2:
$$
\left.\begin{aligned}
x &= 3 - 2t\cr
y &= -2 - 3t\cr
z &= 2 + t\cr\end{aligned}\ \right\}\ t\in\mathbb{R}\qquad\qquad3x - 4y + 2z + 3 = 0
$$
Answer 2:
Priamka a rovina sú rôznobežné.
Question 3:
$$x - 2y + z - 2 = 0\qquad\qquad3x - 4y + 2z + 3 = 0$$
Answer 3:
Roviny sú rôznobežné.
Question 4:
$$
\left.\begin{aligned}
x &=-4+4t\cr
y &= 1+ 3t\cr
z &= -1-2t\end{aligned}\ \right\}\ t\in\mathbb{R}\qquad\qquad 2x - 4y - 2z + 3 = 0$$
Answer 4:
Priamka a rovina sú rovnobežné, pričom priamka neleží v rovine.
Question 5:
$$
\left.\begin{aligned}
x &= 3 -2 t\cr
y &= -2 - 4t\cr
z &= 2 + t\cr
\end{aligned}\ \right\}\ t\in\mathbb{R}\qquad\qquad
\left.\begin{aligned}
x &= -4 + s\cr
y &= -7+2s\cr
z &= 1-2s\cr\end{aligned}\ \right\}\ s\in\mathbb{R}
$$
Answer 5:
Priamky sú mimobežné.
Question 6:
$$
\left.\begin{aligned}
x &= 3+2t\cr
y &= 2+ 3t\cr
z &= 2 + 2t\end{aligned}\ \right\}\ t\in\mathbb{R}\qquad\qquad3x - 4y + 3z - 7 = 0$$
Answer 6:
Priamka leží v rovine.
Question 7:
$$\left.\begin{aligned}
x &= 3 -2t\cr
y &= -2 - 4t\cr
z &= 2 + 6t
\end{aligned}\ \right\}\ t\in\mathbb{R}\qquad\qquad
\left.\begin{aligned}
x &= -4 + s\cr
y &= -7+2s\cr
z &= 1-3s
\end{aligned} \ \right\}\ s\in\mathbb{R}$$
Answer 7:
Priamky sú rovnobežné a rôzne.
Question 8:
$$2x - 6y + 4z - 3 = 0\qquad\qquad 3x - 9y + 6z - 5 = 0$$
Answer 8:
Roviny sú rovnobežné a rôzne.