题目

Problem

With respect to a fixed origin $O$ , the equations of lines $l_1$ and $l_2$ are given by

l_1:\ \mathbf{r}=\begin{pmatrix}2\\8\\10\end{pmatrix}+\lambda\begin{pmatrix}-1\\2\\3\end{pmatrix}

l_2:\ \mathbf{r}=\begin{pmatrix}-4\\-1\\2\end{pmatrix}+\mu\begin{pmatrix}5\\4\\8\end{pmatrix}

where $\lambda$ and $\mu$ are scalar parameters.

Prove that lines $l_1$ and $l_2$ are skew.

(5)

题目中文翻译

相对于固定原点 $O$ ，直线 $l_1$ 和 $l_2$ 的方程分别为

l_1:\ \mathbf{r}=\begin{pmatrix}2\\8\\10\end{pmatrix}+\lambda\begin{pmatrix}-1\\2\\3\end{pmatrix}

l_2:\ \mathbf{r}=\begin{pmatrix}-4\\-1\\2\end{pmatrix}+\mu\begin{pmatrix}5\\4\\8\end{pmatrix}

其中 $\lambda,\mu$ 为标量参数。

证明直线 $l_1$ 与 $l_2$ 是异面直线。

解答

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证明两条直线异面需要完成两件事：先比较方向向量，证明它们不平行；再假设两线相交，联立三个坐标方程。由前两个坐标求出参数后，第三个坐标不相等，因此两线不相交。

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The direction vectors of $l_1$ and $l_2$ are

\begin{pmatrix} -1\\ 2\\ 3 \end{pmatrix} \qquad\text{and}\qquad \begin{pmatrix} 5\\ 4\\ 8 \end{pmatrix}.

They are not scalar multiples, since, for example,

\frac{5}{-1}\ne\frac{4}{2}.

Therefore, $l_1$ and $l_2$ are not parallel.

If the lines intersect, their coordinates must satisfy

\begin{align*} 2-\lambda=&\,-4+5\mu,\\ 8+2\lambda=&\,-1+4\mu,\\ 10+3\lambda=&\,2+8\mu. \end{align*}

Solving the first two equations,

\begin{align*} \lambda+5\mu=&\,6,\\ 2\lambda-4\mu=&\,-9, \end{align*}

which gives

\lambda=-\frac32, \qquad \mu=\frac32.

Substituting these values into the third coordinate,

10+3\bigg(-\frac32\bigg)=\frac{11}{2},

whereas

2+8\bigg(\frac32\bigg)=14.

These values are not equal, so the lines do not intersect.

Since $l_1$ and $l_2$ are neither parallel nor intersecting,

\boxed{l_1\text{ and }l_2\text{ are skew}}.