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 \(a x + b y + c z = d\).

7 Three planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) have equations $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 5 , \quad \mathbf { r } . ( \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 6 , \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } - 12 \mathbf { k } ) = 12 ,$$ 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.
2. Write down what you can deduce about the line of intersection of planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\).
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.

7 Three planes have equations, $$\begin{gathered} x - y + k z = 3 \\ k x - 3 y + 5 z = - 1 \\ x - 2 y + 3 z = - 4 \end{gathered}$$ Where \(k\) is a real constant. The planes do not meet at a unique point. 7

1. Find the possible values of \(k\) 7
2. There are two possible geometric configurations of the given planes. Identify each possible configurations, stating the corresponding value of \(k\) Fully justify your answer.
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3. Given further that the equations of the planes form a consistent system, find the solution of the system of equations.

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 \(l\).