Geometric configuration of planes

10 Two planes \(p\) and \(q\) have equations \(x + y + 3 z = 8\) and \(2 x - 2 y + z = 3\) respectively.

1. Calculate the acute angle between the planes \(p\) and \(q\).
2. The point \(A\) on the line of intersection of \(p\) and \(q\) has \(y\)-coordinate equal to 2 . Find the equation of the plane which contains the point \(A\) and is perpendicular to both the planes \(p\) and \(q\). Give your answer in the form \(a x + b y + c z = d\).

15 The equations of three planes are $$\begin{aligned} - 4 x + k y + 7 z & = 4 \\ x - 2 y + 5 z & = 1 \\ 2 x + 3 y + z & = 2 \end{aligned}$$ Given that the planes form a sheaf, determine the values of \(k\) and \(l\).

Three planes \(\Pi_1\), \(\Pi_2\) and \(\Pi_3\) have equations $$\mathbf{r} \cdot (\mathbf{i} + \mathbf{j} - 2\mathbf{k}) = 5, \quad \mathbf{r} \cdot (\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = 6, \quad \mathbf{r} \cdot (\mathbf{i} + 5\mathbf{j} - 12\mathbf{k}) = 12,$$ respectively. Planes \(\Pi_1\) and \(\Pi_2\) intersect in a line \(l\); planes \(\Pi_2\) and \(\Pi_3\) intersect in a line \(m\).

1. Show that \(l\) and \(m\) are in the same direction. [5]
2. Write down what you can deduce about the line of intersection of planes \(\Pi_1\) and \(\Pi_3\). [1]
3. By considering the cartesian equations of \(\Pi_1\), \(\Pi_2\) and \(\Pi_3\), or otherwise, determine whether or not the three planes have a common line of intersection. [4]