Line intersection with plane

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 \(a x + b y + c z = 1\).

10 The straight line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\). The plane \(p\) has equation \(3 x - y + 2 z = 9\). 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 an equation for the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = d\).

10 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\) and \(5 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) respectively. The plane \(p\) has equation \(x + y = 5\).

1. Find the position vector of the point of intersection of the line through \(A\) and \(B\) and the plane \(p\).
2. A second plane \(q\) has an equation of the form \(x + b y + c z = d\), where \(b , c\) and \(d\) are constants. The plane \(q\) contains the line \(A B\), and the acute angle between the planes \(p\) and \(q\) is \(60 ^ { \circ }\). Find the equation of \(q\).

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

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 .

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 \(l\).

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 \(p\).

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 equation \(\mathbf { r } = 5 \mathbf { i } - 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )\). The plane \(p\) has equation $$( \mathbf { r } - \mathbf { i } - 2 \mathbf { j } ) \cdot ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) = 0$$ The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\).
2. Calculate the acute angle between \(l\) and \(p\).
3. Find the equation of the line which lies in \(p\) and intersects \(l\) at right angles.
   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 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 \(\Pi\).

7. The line \(l _ { 1 }\) has equation $$\frac { x - 3 } { 4 } = \frac { y - 5 } { - 2 } = \frac { z - 4 } { 7 }$$ The plane \(\Pi\) has equation $$2 x + 4 y - z = 1$$ The line \(l _ { 1 }\) intersects the plane \(\Pi\) at the point \(P\)

1. Determine the coordinates of \(P\) The acute angle between \(l _ { 1 }\) and \(\Pi\) is \(\theta\) degrees.
2. Determine, to one decimal place, the value of \(\theta\) The line \(l _ { 2 }\) lies in \(\Pi\) and passes through \(P\)
   Given that the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is also \(\theta\) degrees,
3. determine a vector equation for \(l _ { 2 }\)

1. The line \(l\) passes through the point \(P ( 2,1,3 )\) and is perpendicular to the plane \(\Pi\) whose vector equation is

$$\mathbf { r } . ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) = 3$$ Find

1. a vector equation of the line \(l\),
2. the position vector of the point where \(l\) meets \(\Pi\).
3. Hence find the perpendicular distance of \(P\) from \(\Pi\).

7 The position vectors of the points \(A , B , C , D , G\) are given by $$\mathbf { a } = 6 \mathbf { i } + 4 \mathbf { j } + 8 \mathbf { k } , \quad \mathbf { b } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad \mathbf { c } = \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } , \quad \mathbf { d } = 3 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { g } = 3 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }$$ respectively.

1. The line through \(A\) and \(G\) meets the plane \(B C D\) at \(M\). Write down the vector equation of the line through \(A\) and \(G\) and hence show that the position vector of \(M\) is \(2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k }\).
2. Find the value of the ratio \(A G : A M\).
3. Find the position vector of the point \(P\) on the line through \(C\) and \(G\), such that \(\overrightarrow { C P } = \frac { 4 } { 3 } \overrightarrow { C G }\).
4. Verify that \(P\) lies in the plane \(A B D\).

2 A line \(l\) has equation \(\mathbf { r } = 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + t ( \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } )\) and a plane \(\Pi\) has equation \(8 x - 7 y + 10 z = 7\). Determine whether \(l\) lies in \(\Pi\), is parallel to \(\Pi\) without intersecting it, or intersects \(\Pi\) at one point.

2 In this question, \(L\) is the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2 \\ 2 \\ 1 \end{array} \right)\), and \(\mathrm { g } ( x , y , z ) = \left( x y + z ^ { 2 } \right) \mathrm { e } ^ { x - 2 y }\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
2. Show that the normal to the surface \(\mathrm { g } ( x , y , z ) = 3\) at the point \(( 2,1 , - 1 )\) is the line \(L\). On the line \(L\), there are two points at which \(\mathrm { g } ( x , y , z ) = 0\).
3. Show that one of these points is \(\mathrm { P } ( 0,3,0 )\), and find the coordinates of the other point Q .
4. Show that, if \(x = - 2 \mu , y = 3 + 2 \mu , z = \mu\), and \(\mu\) is small, then $$\mathrm { g } ( x , y , z ) \approx - 6 \mu \mathrm { e } ^ { - 6 }$$ You are given that \(h\) is a small number.
5. There is a point on \(L\), close to P , at which \(\mathrm { g } ( x , y , z ) = h\). Show that this point is approximately $$\left( \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , 3 - \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , - \frac { 1 } { 6 } \mathrm { e } ^ { 6 } h \right)$$
6. Find the approximate coordinates of the point on \(L\), close to Q , at which \(\mathrm { g } ( x , y , z ) = h\).

6 The plane \(\Pi\) has equation \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 6 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ - 3 \\ - 5 \end{array} \right)\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } 7 \\ 4 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 3 \\ 0 \\ - 1 \end{array} \right)\).

1. Express the equation of \(\Pi\) in the form r.n \(= p\).
2. Find the point of intersection of \(l\) and \(\Pi\).
3. The equation of \(\Pi\) may be expressed in the form \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 6 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) + \mu \mathbf { c }\), where \(\mathbf { c }\) is perpendicular to \(\left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right)\). Find \(\mathbf { c }\).

6 The plane \(\Pi\) has equation \(x + 2 y - 2 z = 5\). The line \(l\) has equation \(\frac { x - 1 } { 2 } = \frac { y + 1 } { 5 } = \frac { z - 2 } { 1 }\).

1. Find the coordinates of the point of intersection of \(l\) with the plane \(\Pi\).
2. Calculate the acute angle between \(l\) and \(\Pi\).
3. Find the coordinates of the two points on the line \(l\) such that the distance of each point from the plane \(\Pi\) is 2 .

3 The plane \(\Pi\) passes through the points \(( 1,2,1 ) , ( 2,3,6 )\) and \(( 4 , - 1,2 )\).

1. Find a cartesian equation of the plane \(\Pi\). The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 2 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 4 \\ 3 \\ - 2 \end{array} \right)\).
2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
3. Find the acute angle between \(\Pi\) and \(l\).

3 A line \(l\) has equation \(\frac { x - 6 } { - 4 } = \frac { y + 7 } { 8 } = \frac { z + 10 } { 7 }\) and a plane \(p\) has equation \(3 x - 4 y - 2 z = 8\).

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

6

1. Find the point of intersection of the line \(\mathbf { r } = \left( \begin{array} { r } - 8 \\ - 2 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } - 3 \\ 0 \\ 1 \end{array} \right)\) and the plane \(2 x - 3 y + z = 11\).
2. Find the acute angle between the line and the normal to the plane. Section B (36 marks)

11 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \theta ( 2 \mathbf { j } - \mathbf { k } ) + \phi ( 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )$$ Find a vector normal to \(\Pi _ { 1 }\) and hence show that the equation of \(\Pi _ { 1 }\) can be written as \(2 x + 3 y + 6 z = 14\). The line \(l\) has equation $$\mathbf { r } = 3 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } + t ( 4 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k } )$$ The point on \(l\) where \(t = \lambda\) is denoted by \(P\). Find the set of values of \(\lambda\) for which the perpendicular distance of \(P\) from \(\Pi _ { 1 }\) is not greater than 4 . The plane \(\Pi _ { 2 }\) contains \(l\) and the point with position vector \(\mathbf { i } + 2 \mathbf { j } + \mathbf { k }\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

9 Find a cartesian equation of the plane \(\Pi\) containing the lines $$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - 7 \mathbf { j } + 10 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } )$$ The line \(l\) passes through the point \(P\) with position vector \(6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) and is parallel to the vector \(2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k }\). Find

1. the position vector of the point where \(l\) meets \(\Pi\),
2. the perpendicular distance from \(P\) to \(\Pi\),
3. the acute angle between \(l\) and \(\Pi\).

8 The points \(A , B , C\) have position vectors $$4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k } , \quad 5 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 4 \mathbf { k }$$ respectively, relative to the origin \(O\). Find a cartesian equation of the plane \(A B C\). The point \(D\) has position vector \(6 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }\). Find the coordinates of \(E\), the point of intersection of the line \(O D\) with the plane \(A B C\). Find the acute angle between the line \(E D\) and the plane \(A B C\).

8 A line, passing through the point \(A ( 3,0,2 )\), has vector equation \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\). It meets the plane \(\Pi\), which has equation \(\mathbf { r } \cdot ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) = 3\), at the point \(P\). Find the coordinates of \(P\). Write down a vector \(\mathbf { n }\) which is perpendicular to \(\Pi\), and calculate the vector \(\mathbf { w }\), where $$\mathbf { w } = \mathbf { n } \times ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ The point \(Q\) lies in \(\Pi\) and is the foot of the perpendicular from \(A\) to \(\Pi\). Use the vector \(\mathbf { w }\) to determine an equation of the line \(P Q\) in the form \(\mathbf { r } = \mathbf { u } + \mu \mathbf { v }\).