Line of intersection of planes

9 Two planes have equations \(x + 2 y - 2 z = 2\) and \(2 x - 3 y + 6 z = 3\). The planes intersect in the straight line \(l\).

1. Calculate the acute angle between the two planes.
2. Find a vector equation for the line \(l\).

9 Two planes have equations \(x + 2 y - 2 z = 7\) and \(2 x + y + 3 z = 5\).

1. Calculate the acute angle between the planes.
2. Find a vector equation for the line of intersection of the planes.

9 Two planes have equations \(x + 3 y - 2 z = 4\) and \(2 x + y + 3 z = 5\). The planes intersect in the straight line \(l\).

1. Calculate the acute angle between the two planes.
2. Find a vector equation for the line \(l\).

9 Two planes have equations \(2 x + 3 y - z = 1\) and \(x - 2 y + z = 3\).

1. Find the acute angle between the planes.
2. Find a vector equation for the line of intersection of the planes.

7 Two planes have equations \(2 x - y - 3 z = 7\) and \(x + 2 y + 2 z = 0\).

1. Find the acute angle between the planes.
2. Find a vector equation for their line of intersection.

6 Two planes have equations \(3 x - y + 2 z = 9\) and \(x + y - 4 z = - 1\).

1. Find the acute angle between the planes.
2. Find a vector equation of the line of intersection of the planes.

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

1. Show that the planes are perpendicular.
2. Find a vector equation for the line of intersection of the two planes.

7 The plane \(\Pi _ { 1 }\) has equation \(r = - 4 \mathbf { j } - 3 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( \mathbf { i } + \mathbf { j } - \mathbf { k } )\).

1. Obtain an equation of \(\Pi _ { 1 }\) in the form \(\mathrm { px } + \mathrm { qy } + \mathrm { rz } = \mathrm { d }\).
2. The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( - 5 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) = 4\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   The line \(l\) passes through the point \(A\) with position vector \(a \mathbf { i } + a \mathbf { j } + ( a - 7 ) \mathbf { k }\) and is parallel to \(( 1 - b ) \mathbf { i } + b \mathbf { j } + b \mathbf { k }\), where \(a\) and \(b\) are positive constants.
3. Given that the perpendicular distance from \(A\) to \(\Pi _ { 1 }\) is \(\sqrt { 2 }\), find the value of \(a\).
4. Given that the obtuse angle between \(l\) and \(\Pi _ { 1 }\) is \(\frac { 3 } { 4 } \pi\), find the exact value of \(b\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 The points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } , \quad - \mathbf { j } + \mathbf { k } \quad \text { and } \quad 2 \mathbf { i } + \mathbf { j } - 7 \mathbf { k }$$ respectively, relative to the origin \(O\).

1. Find an equation of the plane \(O A B\), giving your answer in the form \(\mathbf { r } . \mathbf { n } = p\).
   The plane \(\Pi\) has equation \(\mathrm { x } - 3 \mathrm { y } - 2 \mathrm { z } = 1\).
2. Find the perpendicular distance of \(\Pi\) from the origin.
3. Find the acute angle between the planes \(O A B\) and \(\Pi\).
4. Find an equation for the common perpendicular to the lines \(O C\) and \(A B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { k } )\).

1. Find an equation for \(\Pi _ { 1 }\) in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\).
   The line \(l\), which does not lie in \(\Pi _ { 1 }\), has equation \(\mathbf { r } = - 3 \mathbf { i } + \mathbf { k } + t ( \mathbf { i } + \mathbf { j } + \mathbf { k } )\).
2. Show that \(l\) is parallel to \(\Pi _ { 1 }\).
3. Find the distance between \(l\) and \(\Pi _ { 1 }\).
4. The plane \(\Pi _ { 2 }\) has equation \(3 x + 3 y + 2 z = 1\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

7. The plane \(\Pi _ { 1 }\) has equation \(x + y + z = 3\) and the plane \(\Pi _ { 2 }\) has equation \(2 x + 3 y - z = 4\) The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(L\).

1. Find a cartesian equation for the line \(L\). The plane \(\Pi _ { 3 }\) has equation $$\text { r. } \left( \begin{array} { r } 5 \\ - 4 \\ 4 \end{array} \right) = 12$$ The line \(L\) meets the plane \(\Pi _ { 3 }\) at the point \(A\).
2. Find the coordinates of \(A\).
3. Find the acute angle between \(\overrightarrow { O A }\) and the line \(L\), where \(O\) is the origin. Give your answer in degrees to one decimal place.

8. The plane \(\Pi _ { 1 }\) has equation $$x - 5 y + 3 z = 11$$ The plane \(\Pi _ { 2 }\) has equation $$3 x - 2 y + 2 z = 7$$ The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(l\).

1. Find a vector equation for \(l\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\) where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The point \(P ( 2,0,3 )\) lies on \(\Pi _ { 1 }\) The line \(m\), which passes through \(P\), is parallel to \(l\). The point \(Q ( 3,2,1 )\) lies on \(\Pi _ { 2 }\) The line \(n\), which passes through \(Q\), is also parallel to \(l\).
2. Find, in exact simplified form, the shortest distance between \(m\) and \(n\).
   

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1. The plane \(\Pi _ { 1 }\) contains the point \(A ( 2,4 , - 5 )\) and is normal to the vector \(\left( \begin{array} { r } - 1 \\ 3 \\ 3 \end{array} \right)\)

The plane \(\Pi _ { 2 }\) contains the point \(B ( 3,6 , - 2 )\) and is normal to the vector \(\left( \begin{array} { r } 2 \\ 0 \\ - 5 \end{array} \right)\) The line \(l\) is the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)

1. Determine a vector equation for \(l\). The points \(C\) and \(D\) both lie on \(l\).
   Given that \(C\) and \(D\) are 5 units apart,
2. determine the exact volume of the tetrahedron \(A B C D\).

1. The plane \(\Pi _ { 1 }\) has vector equation

$$\mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 0 \end{array} \right) + s \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)$$ where \(s\) and \(t\) are scalar parameters.

1. Determine a Cartesian equation for \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation \(\mathbf { r } . \left( \begin{array} { r } 5 \\ - 2 \\ 3 \end{array} \right) = 1\)
2. Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) Give your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The plane \(\Pi _ { 3 }\) has Cartesian equation \(4 x - 3 y - z = 0\)
3. Use the answer to part (b) to determine the coordinates of the point of intersection of \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\)

8. The plane \(\Pi _ { 1 }\) has vector equation \(\mathbf { r }\). \(\left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) = 5\) The plane \(\Pi _ { 2 }\) has vector equation \(\mathbf { r } . \left( \begin{array} { r } - 1 \\ 2 \\ 4 \end{array} \right) = 7\)

1. Find a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\) where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The plane \(\Pi _ { 3 }\) has cartesian equation $$x - y + 2 z = 31$$
2. Using your answer to part (a), or otherwise, find the coordinates of the point of intersection of the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) \includegraphics[max width=\textwidth, alt={}, center]{393fd7be-c8f5-4b83-a5c7-2de04987a039-16_104_77_2469_1804}

1. The plane \(\Pi _ { 1 }\) has equation

$$x - 5 y - 2 z = 3$$ The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } ) + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(\Pi _ { 1 }\) is perpendicular to \(\Pi _ { 2 }\)
2. Find a cartesian equation for \(\Pi _ { 2 }\)
3. Find an equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = \mathbf { 0 }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors to be found.
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2 Find the equation of the line of intersection of the planes with equations $$\mathbf { r } . ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 4 \quad \text { and } \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } ) = 6 ,$$ giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).

3 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations \(\mathbf { r } \cdot ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) = 1\) and \(\mathbf { r } \cdot ( 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) = 3\) respectively. Find

1. the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), correct to the nearest degree,
2. the equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).

1 Two planes have equations $$x + 2 y + 5 z = 12 \text { and } 2 x - y + 3 z = 5$$

1. Find the acute angle between the planes.
2. Find a vector equation of the line of intersection of the planes.

1

1. Find a vector equation of the line of intersection of the planes \(2 x + y - z = 4\) and \(3 x + 5 y + 2 z = 13\).
2. Find the exact distance of the point \(( 2,5 , - 2 )\) from the plane \(2 x + y - z = 4\).

6 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ - 5 \\ - 2 \end{array} \right)\).

1. Express the equation of \(\Pi _ { 1 }\) in the form r.n \(= p\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . \left( \begin{array} { r } 7 \\ 17 \\ - 3 \end{array} \right) = 21\).
2. Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).

6 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations $$\mathbf { r } \cdot \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) = 3 \text { and } \mathbf { r } \cdot \left( \begin{array} { l } 2 \\ 1 \\ 4 \end{array} \right) = 5$$ respectively. They intersect in the line \(l\).

1. Find cartesian equations of \(l\). The plane \(\Pi _ { 3 }\) has equation \(\mathbf { r } . \left( \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right) = 1\).
2. Show that \(\Pi _ { 3 }\) is parallel to \(l\) but does not contain it.
3. Verify that \(( 2,0,1 )\) lies on planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\). Hence write down a vector equation of the line of intersection of these planes.

9 The plane \(\Pi _ { 1 }\) has parametric equation $$\mathbf { r } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )$$ Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has cartesian equation \(3 x - 2 y - 3 z = 4\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

8 Find a cartesian equation of the plane \(\Pi _ { 1 }\) passing through the points with coordinates \(( 2 , - 1,3 )\), \(( 4,2 , - 5 )\) and \(( - 1,3 , - 2 )\). The plane \(\Pi _ { 2 }\) has cartesian equation \(3 x - y + 2 z = 5\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find a vector equation of the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

9 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), which meet in the line \(/\), have vector equations $$\begin{aligned} & \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 1 } ( 2 \mathbf { i } + 3 \mathbf { k } ) + \phi _ { 1 } ( - 4 \mathbf { j } + 5 \mathbf { k } ) , \\ & \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 2 } ( 3 \mathbf { j } + \mathbf { k } ) + \phi _ { 2 } ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) , \end{aligned}$$ respectively. Find a vector equation of the line \(l\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\). Find a vector equation of the plane \(\Pi _ { 3 }\) which contains \(l\) and which passes through the point with position vector \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\). Find also the equation of \(\Pi _ { 3 }\) in the form \(a x + b y + c z = d\). Deduce, or prove otherwise, that the system of equations $$\begin{aligned} & 6 x - 5 y - 4 z = - 32 \\ & 5 x - y + 3 z = 24 \\ & 9 x - 2 y + 5 z = 40 \end{aligned}$$ has an infinite number of solutions.