4.04b Plane equations: cartesian and vector forms

10 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = \mathbf { i } - 2 \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 6 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } + t ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$ respectively.

1. Show that \(l\) and \(m\) intersect, and find the position vector of their point of intersection.
2. Find the equation of the plane containing \(l\) and \(m\), giving your answer in the form \(a x + b y + c z = d\).

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

1. Show that the line \(l\) lies in the plane \(p\).
2. A second plane is perpendicular to the plane \(p\), parallel to the line \(l\) and contains the point with position vector \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\). Find the equation of this plane, giving your answer in the form \(a x + b y + c z = d\).

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 \(l\).

7 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j }\). The point \(P\) lies on the line \(A B\) and \(O P\) is perpendicular to \(A B\).

1. Find a vector equation for the line \(A B\).
2. Find the position vector of \(P\).
3. Find the equation of the plane which contains \(A B\) and which is perpendicular to the plane \(O A B\), giving your answer in the form \(a x + b y + c z = d\).

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 \(p\).

9 The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } a \\ 1 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 4 \\ 3 \\ - 2 \end{array} \right)\), where \(a\) is a constant. The plane \(p\) has equation \(2 x - 2 y + z = 10\).

1. Given that \(l\) does not lie in \(p\), show that \(l\) is parallel to \(p\).
2. Find the value of \(a\) for which \(l\) lies in \(p\).
3. It is now given that the distance between \(l\) and \(p\) is 6 . Find the possible values of \(a\).

10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 5 \\ - 3 \end{array} \right) .$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).

1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).

10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 5 \\ - 3 \end{array} \right)$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).

1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).

8 Two lines have equations $$\mathbf { r } = \left( \begin{array} { r } 5 \\ 1 \\ - 4 \end{array} \right) + s \left( \begin{array} { r } 1 \\ - 1 \\ 3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } p \\ 4 \\ - 2 \end{array} \right) + t \left( \begin{array} { r } 2 \\ 5 \\ - 4 \end{array} \right) ,$$ where \(p\) is a constant. It is given that the lines intersect.

1. Find the value of \(p\) and determine the coordinates of the point of intersection.
2. Find the equation of the plane containing the two lines, giving your answer in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers.

9
The diagram shows three points \(A , B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 0 \\ 3 \\ 1 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 3 \\ 0 \\ 4 \end{array} \right)\). The point \(D\) lies on \(B C\), between \(B\) and \(C\), and is such that \(C D = 2 D B\).

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 position vector of \(D\).
3. Show that the length of the perpendicular from \(A\) to \(O D\) is \(\frac { 1 } { 3 } \sqrt { } ( 65 )\).

10 The line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } - 9 \mathbf { j } + 9 \mathbf { k } + \lambda ( - 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\). The point \(A\) has position vector \(3 \mathbf { i } + 8 \mathbf { j } + 5 \mathbf { k }\).

1. Show that the length of the perpendicular from \(A\) to \(l\) is 15 .
2. The line \(l\) lies in the plane with equation \(a x + b y - 3 z + 1 = 0\), where \(a\) and \(b\) are constants. Find the values of \(a\) and \(b\).

7 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right)$$ The plane \(m\) is perpendicular to \(A B\) and contains the point \(C\).

1. Find a vector equation for the line passing through \(A\) and \(B\).
2. Obtain the equation of the plane \(m\), giving your answer in the form \(a x + b y + c z = d\).
3. The line through \(A\) and \(B\) intersects the plane \(m\) at the point \(N\). Find the position vector of \(N\) and show that \(C N = \sqrt { } ( 13 )\).

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 \(C\).

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.

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.

7 The plane \(m\) has equation \(x + 4 y - 8 z = 2\). The plane \(n\) is parallel to \(m\) and passes through the point \(P\) with coordinates \(( 5,2 , - 2 )\).

1. Find the equation of \(n\), giving your answer in the form \(a x + b y + c z = d\).
2. Calculate the perpendicular distance between \(m\) and \(n\).
3. The line \(l\) lies in the plane \(n\), passes through the point \(P\) and is perpendicular to \(O P\), where \(O\) is the origin. Find a vector equation for \(l\).

7 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right)$$ The plane \(m\) is perpendicular to \(A B\) and contains the point \(C\).

1. Find a vector equation for the line passing through \(A\) and \(B\).
2. Obtain the equation of the plane \(m\), giving your answer in the form \(a x + b y + c z = d\).
3. The line through \(A\) and \(B\) intersects the plane \(m\) at the point \(N\). Find the position vector of \(N\) and show that \(C N = \sqrt { } ( 13 )\).

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 \(\Pi\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = - 5 \mathbf { j } + \lambda ( 5 \mathbf { i } + 2 \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { j } + \mathbf { k } )\) respectively. The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).

1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the distance between \(l _ { 2 }\) and \(\Pi\).
   The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).
3. Show that \(P\) has position vector \(\frac { 55 } { 27 } \mathbf { i } - 5 \mathbf { j } + \frac { 22 } { 27 } \mathbf { k }\) and state a vector equation for \(P Q\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 \(\Pi _ { 2 }\).

6 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = - \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + s ( 2 \mathbf { i } - 3 \mathbf { j } )\) and \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { k } + t ( 3 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) respectively. The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and the point \(P\) with position vector \(- 2 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }\).

1. Find an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
   The plane \(\Pi _ { 2 }\) contains \(l _ { 2 }\) and is parallel to \(l _ { 1 }\).
2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
4. The point \(Q\) is such that \(\overrightarrow { \mathrm { OQ } } = - 5 \overrightarrow { \mathrm { OP } }\). Find the position vector of the foot of the perpendicular from the point \(Q\) to \(\Pi _ { 2 }\).