Project ID:
7300000044
Accepted:
Type:
Layout:
Question:
Let the $x$-axis, $y$-axis, and $z$-axis be the axes of a three-dimensional coordinate system. Let $p$ denote a straight line. Match each given parametric expression of the line $p$ with its correct position relative to the coordinate axes.
Questions Title:
Parametric expression of the line $p$
Answers Title:
Relative position of the line $p$ to the coordinate axes
Question 1:
$$\left.\begin{aligned}
x &= 2 - 2t\cr
y &= -3\cr
z &= 4\end{aligned}
\ \right\}
\ t\in \mathbb{R}$$
Answer 1:
The line $p$ is parallel to the $x$-axis.
Question 2:
$$
\left.
\begin{aligned}
x &= 2\cr
y &= -3+3t\cr
z &= 4\ \end{aligned}\right\}\ t\in\mathbb{R}$$
Answer 2:
The line $p$ is parallel to the $y$-axis.
Question 3:
$$
\left.
\begin{aligned}
x &= 2\cr
y &= -3\cr
z &= 4-4t\end{aligned}\ \right\}\ t\in\mathbb{R}$$
Answer 3:
The line $p$ is parallel to the $z$-axis.
Question 4:
$$\left.\begin{aligned}
x &= 2-t\cr
y &= -3+3t\cr
z &= 4-4t\end{aligned}\ \right\}\ t\in\mathbb{R}$$
Answer 4:
The line $p$ intersects the $x$-axis only.
Question 5:
$$
\left.
\begin{aligned}
x &= 2-2t\cr
y &= -3+t\cr
z &= 4-4t
\end{aligned} \ \right\} \ t \in \mathbb{R}
$$
Answer 5:
The line $p$ intersects the $y$-axis only.
Question 6:
$$\left.\begin{aligned}
x &= 2-2t\cr
y &= -3+3t\cr
z &= 4-t\end{aligned}\ \right\}\ t\in \mathbb{R}$$
Answer 6:
The line $p$ intersects the $z$-axis only.
Question 7:
$$\left.\begin{aligned}
x &= 2-2t\cr
y &= -3+3t\cr
z &= 4-4t\end{aligned}\ \right\}\ t\in \mathbb{R}$$
Answer 7:
The line $p$ intersects all coordinate axes.
Question 8:
$$\left.\begin{aligned}
x &= 2-t\cr
y &= -3+t\cr
z &= 4-t\end{aligned}\ \right\}\ t\in \mathbb{R}$$
Answer 8:
The line $p$ is skew to all coordinate axes.