Parallel and perpendicular planes

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 \(A B\).

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]

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 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 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 \(l\).

2 Write down normal vectors to the planes \(2 x + 3 y + 4 z = 10\) and \(x - 2 y + z = 5\).
Hence show that these planes are perpendicular to each other.

2 Two intersecting lines, lying in a plane \(p\), have equations $$\frac { x - 1 } { 2 } = \frac { y - 3 } { 1 } = \frac { z - 4 } { - 3 } \quad \text { and } \quad \frac { x - 1 } { - 1 } = \frac { y - 3 } { 2 } = \frac { z - 4 } { 4 } .$$

1. Obtain the equation of \(p\) in the form \(2 x - y + z = 3\).
2. Plane \(q\) has equation \(2 x - y + z = 21\). Find the distance between \(p\) and \(q\).

10 The coordinates of the points \(A\) and \(B\) are ( \(3 , - 2 , - 1\) ) and ( \(13,10,9\) ) respectively.

• The plane \(\Pi _ { A }\) contains \(A\) and the plane \(\Pi _ { B }\) contains \(B\).
• The planes \(\Pi _ { A }\) and \(\Pi _ { B }\) are parallel.
• The \(x\) and \(y\) components of any normal to plane \(\Pi _ { A }\) are equal.
• The shortest distance between \(\Pi _ { A }\) and \(\Pi _ { B }\) is 2 .

There are two possible solution planes for \(\Pi _ { A }\) which satisfy the above conditions.
Determine the acute angle between these two possible solution planes.

2

1. Show that the vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\) is parallel to the plane \(2 \mathrm { x } + \mathrm { y } - 3 \mathrm { z } = 10\).
2. Determine the acute angle between the planes \(2 x + y - 3 z = 10\) and \(x - y - 3 z = 3\).

10 A vector \(\mathbf { v }\) has magnitude 1 unit. The angle between \(\mathbf { v }\) and the positive \(z\)-axis is \(60 ^ { \circ }\), and \(\mathbf { v }\) is parallel to the plane \(x - 2 y = 0\). Given that \(\mathbf { v } = a \mathbf { i } + b \mathbf { j } + c \mathbf { k }\), where \(a , b\) and \(c\) are all positive, find \(\mathbf { v }\). \section*{END OF QUESTION PAPER}

7 The plane \(\Pi\) has equation \(3 x - 5 y + z = 9\).

1. Show that \(\Pi\) contains
   • the point (4,1,2)
     and
   • the vector \(\left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)\).
   • Determine the equation of a plane which is perpendicular to \(\Pi\) and which passes through ( \(4,1,2\) ).

2 The plane \(x + 2 y + c z = 4\) is perpendicular to the plane \(2 x - c y + 6 z = 9\), where \(c\) is a constant. Find the value of \(c\).

9 In this question you must show detailed reasoning. Find a vector \(\mathbf { v }\) which has the following properties.

• It is a unit vector.
• It is parallel to the plane \(2 x + 2 y + z = 10\).
• It makes an angle of \(45 ^ { \circ }\) with the normal to the plane \(\mathrm { x } + \mathrm { z } = 5\).

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