4.04d Angles: between planes and between line and plane

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 \includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-3_693_537_1206_804} The diagram shows a set of rectangular axes \(O x , O y\) and \(O z\), and three points \(A , B\) and \(C\) with position vectors \(\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 0 \\ 0 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Calculate the acute angle between the planes \(A B C\) and \(O A B\).

3 Points \(A\) and \(B\) have coordinates \(( - 1,2,5 )\) and \(( 2 , - 2,11 )\) respectively. The plane \(p\) passes through \(B\) and is perpendicular to \(A B\).

1. Find an equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the acute angle between \(p\) and the \(y\)-axis.

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.

10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( a \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\), where \(a\) is a constant. The plane \(p\) has equation \(x + 2 y + 2 z = 6\). Find the value or values of \(a\) in each of the following cases.

1. The line \(l\) is parallel to the plane \(p\).
2. The line \(l\) intersects the line passing through the points with position vectors \(3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and \(\mathbf { i } + \mathbf { j } - \mathbf { k }\).
3. The acute angle between the line \(l\) and the plane \(p\) is \(\tan ^ { - 1 } 2\).

10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } + \lambda ( 3 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )\) and the plane \(p\) has equation \(2 x + 3 y - 5 z = 18\).

1. Find the position vector of the point of intersection of \(l\) and \(p\).
2. Find the acute angle between \(l\) and \(p\).
3. A second plane \(q\) is perpendicular to the plane \(p\) and contains the line \(l\). Find the equation of \(q\), giving your answer in the form \(a x + b y + c z = d\).

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.

10 The straight line \(l\) passes through the points \(A\) and \(B\) with position vectors $$2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$$ respectively. This line intersects the plane \(p\) with equation \(x - 2 y + 2 z = 6\) at the point \(C\).

1. Find the position vector of \(C\).
2. Find the acute angle between \(l\) and \(p\).
3. Show that the perpendicular distance from \(A\) to \(p\) is equal to 2 .

9

1. Express \(\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

10 The straight line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 6 \mathbf { j } - 3 \mathbf { k } + s ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )\). The plane \(p\) has equation \(( \mathbf { r } - 3 \mathbf { i } ) \cdot ( 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k } ) = 0\). The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\).
2. Find the acute angle between \(l\) and \(p\).
3. Find a vector equation for the line which lies in \(p\), passes through \(A\) and is perpendicular to \(l\).

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.

10 The plane \(p\) has equation \(2 x - 3 y + 6 z = 16\). The plane \(q\) is parallel to \(p\) and contains the point with position vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\).

1. Find the equation of \(q\), giving your answer in the form \(a x + b y + c z = d\).
2. Calculate the perpendicular distance between \(p\) and \(q\).
3. The line \(l\) is parallel to the plane \(p\) and also parallel to the plane with equation \(x - 2 y + 2 z = 5\). Given that \(l\) passes through the origin, find a vector equation for \(l\).

6 The straight line \(l\) passes through the points with coordinates \(( - 5,3,6 )\) and \(( 5,8,1 )\). The plane \(p\) has equation \(2 x - y + 4 z = 9\).

1. Find the coordinates of the point of intersection of \(l\) and \(p\).
2. Find the acute angle between \(l\) and \(p\).

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 A plane has equation \(4 x - y + 5 z = 39\). A straight line is parallel to the vector \(\mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }\) and passes through the point \(A ( 0,2 , - 8 )\). The line meets the plane at the point \(B\).

1. Find the coordinates of \(B\).
2. Find the acute angle between the line and the plane.
3. The point \(C\) lies on the line and is such that the distance between \(C\) and \(B\) is twice the distance between \(A\) and \(B\). Find the coordinates of each of the possible positions of the point \(C\).

10 The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\).

1. Find the position vectors of the two points on the line whose distance from the origin is \(\sqrt { } ( 10 )\).
2. The plane \(p\) has equation \(a x + y + z = 5\), where \(a\) is a constant. The acute angle between the line \(l\) and the plane \(p\) is equal to \(\sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)\). Find the possible values of \(a\).

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

10 The planes \(m\) and \(n\) have equations \(3 x + y - 2 z = 10\) and \(x - 2 y + 2 z = 5\) respectively. The line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).

1. Show that \(l\) is parallel to \(m\).
2. Calculate the acute angle between the planes \(m\) and \(n\).
3. A point \(P\) lies on the line \(l\). The perpendicular distance of \(P\) from the plane \(n\) is equal to 2 . Find the position vectors of the two possible positions of \(P\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )\). The plane \(p\) has equation \(2 x + y - 3 z = 5\).

1. Find the position vector of the point of intersection of \(l\) and \(p\).
2. Calculate the acute angle between \(l\) and \(p\).
3. A second plane \(q\) is perpendicular to the plane \(p\) and contains the line \(l\). Find the equation of \(q\), giving your answer in the form \(a x + b y + c z = d\).
   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 lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 3 \mathbf { i } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k } )\) and \(\mathbf { r } = 3 \mathbf { i } - 5 \mathbf { j } - 6 \mathbf { k } + \mu ( 5 \mathbf { j } + 6 \mathbf { k } )\) respectively.

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to the vector \(\mathbf { i } + \mathbf { k }\).
2. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
3. Find the acute angle between \(l _ { 2 }\) and \(\Pi\).

6 Let \(t\) be a positive constant.
The line \(l _ { 1 }\) passes through the point with position vector \(t \mathbf { i } + \mathbf { j }\) and is parallel to the vector \(- 2 \mathbf { i } - \mathbf { j }\). The line \(l _ { 2 }\) passes through the point with position vector \(\mathbf { j } + t \mathbf { k }\) and is parallel to the vector \(- 2 \mathbf { j } + \mathbf { k }\). It is given that the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\) is \(\sqrt { \mathbf { 2 1 } }\).

1. Find the value of \(t\).
   The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
2. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
   The plane \(\Pi _ { 2 }\) has Cartesian equation \(5 x - 6 y + 7 z = 0\).
3. Find the acute angle between \(l _ { 2 }\) and \(\Pi _ { 2 }\).
4. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

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.

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

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from \(O\) to the plane \(A B C\).
3. Find the acute angle between the planes \(O A B\) and \(A B C\).

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

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the acute angle between the planes \(O B C\) and \(A B C\).
   The point \(D\) has position vector \(t \mathbf { i } - \mathbf { j }\).
3. Given that the shortest distance between the lines \(A B\) and \(C D\) is \(\sqrt { \mathbf { 1 0 } }\), find the value of \(t\).
   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 \(\Pi\) has equation \(\mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { k } ) + \mu ( 2 \mathbf { i } + 3 \mathbf { j } )\).

1. Find a Cartesian equation of \(\Pi\), giving your answer in the form \(a x + b y + c = d\).
   The line \(l\) passes through the point \(P\) with position vector \(2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\) and is parallel to the vector \(\mathbf { k }\).
2. Find the position vector of the point where \(l\) meets \(\Pi\).
3. Find the acute angle between \(l\) and \(\Pi\).
4. Find the perpendicular distance from \(P\) to \(\Pi\).