向量：练习题

Find the vector equation of the common perpendicular to the skew lines \(L_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + t \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}\) and \(L_2: \mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\).

Find the acute angle, correct to one decimal place, between the plane \(\Pi: x - 2y + 2z = 5\) and the line \(L: \mathbf{r} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}\).

Find the Cartesian equation of the plane that passes through the point \(A(1, 2, 3))\) and contains the line \(\mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix} + t \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\).

The line \(L\) has equation \(\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\). The plane \(\Pi\) has equation \(x - y + 2z = 5\). Find the acute angle between the line \(L\) and the plane \(\Pi\).

A plane has the Cartesian equation \(2x - y + 3z = 7\).
(a) Write down a normal vector to the plane.
(b) Determine whether the point \(A(1, 2, 3)\) lies on the plane.
(c) Find the perpendicular distance from the origin to the plane.

A plane \(\Pi\) has the vector equation \(\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}\).
(a) Find the Cartesian equation of the plane \(\Pi\).
(b) Calculate the perpendicular distance from the point \(P(4, 5, 1)\) to the plane \(\Pi\).