4.04f Line-plane intersection: find point

8 The straight line \(l\) passes through the points \(A\) and \(B\) whose position vectors are \(\mathbf { i } + \mathbf { k }\) and \(4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) respectively. The plane \(p\) has equation \(x + 3 y - 2 z = 3\).

1. Given that \(l\) intersects \(p\), find the position vector of the point of intersection.
2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = 1\).

9 The line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } + t ( 2 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )\). It is given that \(l\) lies in the plane with equation \(2 x + b y + c z = 1\), where \(b\) and \(c\) are constants.

1. Find the values of \(b\) and \(c\).
2. The point \(P\) has position vector \(2 \mathbf { j } + 4 \mathbf { k }\). Show that the perpendicular distance from \(P\) to \(l\) is \(\sqrt { } 5\).

9 The plane \(p\) has equation \(3 x + 2 y + 4 z = 13\). A second plane \(q\) is perpendicular to \(p\) and has equation \(a x + y + z = 4\), where \(a\) is a constant.

1. Find the value of \(a\).
2. The line with equation \(\mathbf { r } = \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) meets the plane \(p\) at the point \(A\) and the plane \(q\) at the point \(B\). Find the length of \(A B\).

8 The point \(P\) has coordinates \(( - 1,4,11 )\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 3 \\ - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)\).

1. Find the perpendicular distance from \(P\) to \(l\).
2. Find the equation of the plane which contains \(P\) and \(l\), giving your answer in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers.

10 Two planes, \(m\) and \(n\), have equations \(x + 2 y - 2 z = 1\) and \(2 x - 2 y + z = 7\) respectively. The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } - \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).

1. Show that \(l\) is parallel to \(m\).
2. Find the position vector of the point of intersection of \(l\) and \(n\).
3. A point \(P\) lying on \(l\) is such that its perpendicular distances from \(m\) and \(n\) are equal. Find the position vectors of the two possible positions for \(P\) and calculate the distance between them.
   [0pt] [The perpendicular distance of a point with position vector \(x _ { 1 } \mathbf { i } + y _ { 1 } \mathbf { j } + z _ { 1 } \mathbf { k }\) from the plane \(a x + b y + c z = d\) is \(\frac { \left| a x _ { 1 } + b y _ { 1 } + c z _ { 1 } - d \right| } { \sqrt { } \left( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } \right) }\).]

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 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } + \mathbf { j } + \mathbf { k }\) respectively. The line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. The plane \(m\) is perpendicular to \(A B\) and passes through the mid-point of \(A B\). The plane \(m\) intersects the line \(l\) at the point \(P\). Find the equation of \(m\) and the position vector of \(P\).

6 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\). The plane \(p\) has equation \(3 x + y - 5 z = 20\).

1. Show that the line \(l\) lies in the plane \(p\).
2. A second plane is parallel to \(l\), perpendicular to \(p\) and contains the point with position vector \(3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\). Find the equation of this plane, giving your answer in the form \(a x + b y + c z = d\). [5]

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 .

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 \(a\).

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.

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 \(C\).

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 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 \(C N = \sqrt { } ( 13 )\).

2 The points \(A , B , C\) have position vectors $$4 \mathbf { i } - 4 \mathbf { j } + \mathbf { k } , \quad - 4 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } , \quad 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ,$$ 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. The point \(D\) has position vector \(2 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k }\). Find the coordinates of the point of intersection of the line \(O D\) with the plane \(A B C\).

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 \(\Pi\).

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 \(\Pi _ { 2 }\).

9 Let \(\mathrm { f } ( x ) = \frac { 2 + 11 x - 10 x ^ { 2 } } { ( 1 + 2 x ) ( 1 - 2 x ) ( 2 + x ) }\).

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

8. The line \(l\) has equation $$\mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) , \text { where } \lambda \text { is a scalar parameter, }$$ and the plane \(\Pi\) has equation $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 19$$

1. Find the coordinates of the point of intersection of \(l\) and \(\Pi\). The perpendicular to \(\Pi\) from the point \(A ( 2,1 , - 2 )\) meets \(\Pi\) at the point \(B\).
2. Verify that the coordinates of \(B\) are \(( 4,3 , - 6 )\). The point \(A ( 2,1 , - 2 )\) is reflected in the plane \(\Pi\) to give the image point \(A ^ { \prime }\).
3. Find the coordinates of the point \(A ^ { \prime }\).
4. Find an equation for the line obtained by reflecting the line \(l\) in the plane \(\Pi\), giving your answer in the form $$\mathbf { r } \times \mathbf { a } = \mathbf { b } ,$$ where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.

9 The lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } - \mathbf { k } )\) respectively, where \(a\) and \(b\) are constants.

1. Given that \(l\) and \(m\) intersect, show that $$2 a - b = 4 .$$
2. Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\).
3. When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\).

1. The plane \(\Pi\) has equation

$$3 x + 4 y - z = 17$$ The line \(l _ { 1 }\) is perpendicular to \(\Pi\) and passes through the point \(P ( - 4 , - 5,3 )\) The line \(l _ { 1 }\) intersects \(\Pi\) at the point \(Q\)

1. Determine the coordinates of \(Q\) Given that the point \(R ( - 1,6,4 )\) lies on \(\Pi\)
2. determine a Cartesian equation of the plane containing \(P Q R\) The line \(l _ { 2 }\) passes through \(P\) and \(R\) The line \(l _ { 3 }\) is the reflection of \(l _ { 2 }\) in \(\Pi\)
3. Determine a vector equation for \(l _ { 3 }\)

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 \(\Pi _ { 3 }\)