Line intersection with plane

8

1. Show that the line \(l\) with vector equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 5 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 5 \\ - 2 \\ 3 \end{array} \right)\) lies in the plane \(\Pi\) with cartesian equation \(x + 4 y + z + 11 = 0\).
2. The plane \(\Pi\) is horizontal, and the point \(P ( 1,2 , k )\) is above it. Given that the point in \(\Pi\) which is directly beneath \(P\) is on the line \(l\), determine the value of \(k\).

A line \(l\) has equation \(\mathbf{r} = 3\mathbf{i} + \mathbf{j} - 2\mathbf{k} + t(\mathbf{i} + 4\mathbf{j} + 2\mathbf{k})\) and a plane \(\Pi\) has equation \(8x - 7y + 10z = 7\). Determine whether \(l\) lies in \(\Pi\), is parallel to \(\Pi\) without intersecting it, or intersects \(\Pi\) at one point. [5]

The line \(L\) passes through the points A\((1, 2, 3)\) and B\((2, 3, 1)\).

1. 1. Find the vector \(\overrightarrow{AB}\).
   2. Write down the vector equation of the line \(L\). [3]
2. The plane \(\Pi\) has equation \(x + 3y - 2z = 5\).
   1. Find the coordinates of the point of intersection of \(L\) and \(\Pi\).
   2. Find the acute angle between \(L\) and \(\Pi\). [9]

The equations of two intersecting lines \(l_1\) and \(l_2\) are $$l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ a \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 7 \\ 9 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}$$ where \(a\) is a constant. The equation of the plane \(\Pi\) is $$\mathbf{r} \cdot \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix} = -14.$$ \(l_1\) and \(\Pi\) intersect at \(Q\). \(l_2\) and \(\Pi\) intersect at \(R\).

1. Verify that the coordinates of \(R\) are \((13, 3, -14)\). [2]
2. Determine the exact value of the length of \(QR\). [7]