4.04b Plane equations: cartesian and vector forms

1. The line \(l\) has equation

$$\frac { x + 2 } { 1 } = \frac { y - 5 } { - 1 } = \frac { z - 4 } { - 3 }$$ The plane \(\Pi\) has equation $$\mathbf { r } . ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) = - 7$$ Determine whether the line \(l\) intersects \(\Pi\) at a single point, or lies in \(\Pi\), or is parallel to \(\Pi\) without intersecting it.
(5)

1. All units in this question are in metres.

A lawn is modelled as a plane that contains the points \(L ( - 2 , - 3 , - 1 ) , M ( 6 , - 2,0 )\) and \(N ( 2,0,0 )\), relative to a fixed origin \(O\).

1. Determine a vector equation of the plane that models the lawn, giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\)
2. 1. Show that, according to the model, the lawn is perpendicular to the vector \(\left( \begin{array} { c } 1 \\ 2 \\ - 10 \end{array} \right)\)
   2. Hence determine a Cartesian equation of the plane that models the lawn. There are two posts set in the lawn.
      There is a washing line between the two posts.
      The washing line is modelled as a straight line through points at the top of each post with coordinates \(P ( - 10,8,2 )\) and \(Q ( 6,4,3 )\).
3. Determine a vector equation of the line that models the washing line.
4. State a limitation of one of the models. The point \(R ( 2,5,2.75 )\) lies on the washing line.
5. Determine, according to the model, the shortest distance from the point \(R\) to the lawn, giving your answer to the nearest cm. Given that the shortest distance from the point \(R\) to the lawn is actually 1.5 m ,
6. use your answer to part (e) to evaluate the model, explaining your reasoning.

1. A mining company has identified a mineral layer below ground.

The mining company wishes to drill down to reach the mineral layer and models the situation as follows. With respect to a fixed origin \(O\),

• the ground is modelled as a horizontal plane with equation \(z = 0\)
• the mineral layer is modelled as part of the plane containing the points \(A ( 10,5 , - 50 ) , B ( 15,30 , - 45 )\) and \(C ( - 5,20 , - 60 )\), where the units are in metres
  1. Determine an equation for the plane containing \(A , B\) and \(C\), giving your answer in the form r.n \(= d\)
  2. Determine, according to the model, the acute angle between the ground and the plane containing the mineral layer. Give your answer to the nearest degree.

The mining company plans to drill vertically downwards from the point \(( 5,12,0 )\) on the ground to reach the mineral layer.

Using the model, determine, in metres to 1 decimal place, the distance the mining company will need to drill in order to reach the mineral layer.

State a limitation of the assumption that the mineral layer can be modelled as a plane.

1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 2 \\ 2 \\ 0 \end{array} \right) + \lambda \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right)\) where \(\lambda\) is a scalar parameter.

The line \(l _ { 2 }\) is parallel to \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\)

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular. The plane \(\Pi\) contains the line \(l _ { 1 }\) and is perpendicular to \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\)
2. Determine a Cartesian equation of \(\Pi\)
3. Verify that the point \(A ( 3,1,1 )\) lies on \(\Pi\) Given that
   • the point of intersection of \(\Pi\) and \(l _ { 2 }\) has coordinates \(( 2,3,2 )\)
   • the point \(B ( p , q , r )\) lies on \(l _ { 2 }\)
   • the distance \(A B\) is \(2 \sqrt { 5 }\)
   • \(p , q\) and \(r\) are positive integers
   • determine the coordinates of \(B\).

1. The plane \(\Pi\) passes through the point \(A\) and is perpendicular to the vector \(\mathbf { n }\)

Given that $$\overrightarrow { O A } = \left( \begin{array} { r } 5 \\ - 3 \\ - 4 \end{array} \right) \quad \text { and } \quad \mathbf { n } = \left( \begin{array} { r } 3 \\ - 1 \\ 2 \end{array} \right)$$ where \(O\) is the origin,

1. find a Cartesian equation of \(\Pi\). With respect to the fixed origin \(O\), the line \(l\) is given by the equation $$\mathbf { r } = \left( \begin{array} { r } 7 \\ 3 \\ - 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ - 5 \\ 3 \end{array} \right)$$ The line \(l\) intersects the plane \(\Pi\) at the point \(X\).
2. Show that the acute angle between the plane \(\Pi\) and the line \(l\) is \(21.2 ^ { \circ }\) correct to one decimal place.
3. Find the coordinates of the point \(X\).

1. The line \(l _ { 1 }\) has equation

$$\frac { x - 1 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z - 4 } { 3 }$$ The line \(l _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 3 \mathbf { k } + t ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where \(t\) is a scalar parameter.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) lie in the same plane.
2. Write down a vector equation for the plane containing \(l _ { 1 }\) and \(l _ { 2 }\)
3. Find, to the nearest degree, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\)

1. The plane \(\Pi _ { 1 }\) has equation

$$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Find a Cartesian equation for \(\Pi _ { 1 }\) The line \(l\) has equation $$\frac { x - 1 } { 5 } = \frac { y - 3 } { - 3 } = \frac { z + 2 } { 4 }$$
2. Find the coordinates of the point of intersection of \(l\) with \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r . } ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 5$$
3. Find, to the nearest degree, the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)

5 The line \(l _ { 1 }\) has equation \(\frac { x + 5 } { 1 } = \frac { y + 4 } { - 3 } = \frac { z - 3 } { 5 }\) The plane \(\Pi _ { 1 }\) has equation \(2 x + 3 y - 2 z = 6\)

1. Find the point of intersection of \(l _ { 1 }\) and \(\Pi _ { 1 }\) The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi _ { 1 }\)
2. Show that a vector equation for the line \(l _ { 2 }\) is $$\mathbf { r } = \left( \begin{array} { r } - 7 \\ 2 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ 6 \\ 2 \end{array} \right)$$ where \(\mu\) is a scalar parameter. The plane \(\Pi _ { 2 }\) contains the line \(l _ { 1 }\) and the line \(l _ { 2 }\)
3. Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The plane \(\Pi _ { 3 }\) has equation r. \(\left( \begin{array} { l } 1 \\ 1 \\ a \end{array} \right) = b\) where \(a\) and \(b\) are constants.
   Given that the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) form a sheaf,
4. determine the value of \(a\) and the value of \(b\).

1. The line \(l _ { 1 }\) has equation

$$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathbf { r } = 5 \mathbf { i } + p \mathbf { j } - 7 \mathbf { k } + \mu ( 6 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant.
The plane \(\Pi\) contains \(l _ { 1 }\) and \(l _ { 2 }\)

1. Show that the vector \(3 \mathbf { i } - 10 \mathbf { j } - \mathbf { k }\) is perpendicular to \(\Pi\)
2. Hence determine a Cartesian equation of \(\Pi\)
3. Hence determine the value of \(p\) Given that
   • the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\)
   • the point \(B\) has coordinates \(( 12 , - 11,6 )\)
   • determine, to the nearest degree, the acute angle between \(A B\) and \(\Pi\)

1. The line \(l _ { 1 }\) has equation \(\frac { x - 2 } { 4 } = \frac { y - 4 } { - 2 } = \frac { z + 6 } { 1 }\)

The plane \(\Pi\) has equation \(x - 2 y + z = 6\) The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi\).
Find a vector equation of the line \(l _ { 2 }\)

1. With respect to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates \(( 3,4,5 ) , ( 10 , - 1,5 )\) and ( \(4,7 , - 9\) ) respectively.

The plane \(\Pi\) has equation \(4 x - 8 y + z = 2\) The line segment \(A B\) meets the plane \(\Pi\) at the point \(P\) and the line segment \(B C\) meets the plane \(\Pi\) at the point \(Q\).

1. Show that, to 3 significant figures, the area of quadrilateral \(A P Q C\) is 38.5 The point \(D\) has coordinates \(( k , 4 , - 1 )\), where \(k\) is a constant.
   Given that the vectors \(\overrightarrow { A B } , \overrightarrow { A C }\) and \(\overrightarrow { A D }\) form three edges of a parallelepiped of volume 226
2. find the possible values of the constant \(k\).

The points \(A , B\) and \(C\), with position vectors \(\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \mathbf { b } = \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }\) and \(\mathbf { c } = - 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\) respectively, lie on the plane \(\Pi\)

1. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\)
2. Find an equation for \(\Pi\) in the form r.n \(= p\) The point \(D\) has position vector \(8 \mathbf { i } + 7 \mathbf { j } + 5 \mathbf { k }\)
3. Determine the volume of the tetrahedron \(A B C D\)

The points \(P , Q\) and \(R\) have position vectors \(\left( \begin{array} { r } 1 \\ - 2 \\ 4 \end{array} \right) , \left( \begin{array} { r } 3 \\ 1 \\ - 5 \end{array} \right)\) and \(\left( \begin{array} { l } 2 \\ 0 \\ 3 \end{array} \right)\) respectively.

1. Determine a vector equation of the plane that passes through the points \(P , Q\) and \(R\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\), where \(\lambda\) and \(\mu\) are scalar parameters.
2. Determine the coordinates of the point of intersection of the plane with the \(x\)-axis.

1. A tetrahedron has vertices \(A ( 1,2,1 ) , B ( 0,1,0 ) , C ( 2,1,3 )\) and \(D ( 10,5,5 )\).

Find

1. a Cartesian equation of the plane \(A B C\).
2. the volume of the tetrahedron \(A B C D\). The plane \(\Pi\) has equation \(2 x - 3 y + 3 = 0\) The point \(E\) lies on the line \(A C\) and the point \(F\) lies on the line \(A D\).
   Given that \(\Pi\) contains the point \(B\), the point \(E\) and the point \(F\),
3. find the value of \(k\) such that \(\overrightarrow { A E } = k \overrightarrow { A C }\). Given that \(\overrightarrow { A F } = \frac { 1 } { 9 } \overrightarrow { A D }\)
4. show that the volume of the tetrahedron \(A B C D\) is 45 times the volume of the tetrahedron \(A B E F\).

4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 7 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z - 6 } { 3 }\) and \(\frac { x - 2 } { 1 } = \frac { y - 6 } { 2 } = \frac { z + 2 } { 1 }\) respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
2. Find the cartesian equation of the plane that contains \(l _ { 1 }\) and \(l _ { 2 }\).

5 Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), have equations \(3 x + 2 y + z = 4\) and \(2 x + y + z = 3\) respectively.

1. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). The line \(L\) has equation \(x = 1 - y = 2 - z\).
2. Show that \(L\) lies in both planes.

1 Points \(A , B\) and \(C\) have coordinates \(( 0,1 , - 4 ) , ( 1,1 , - 2 )\) and \(( 3,2,5 )\) respectively.

1. Find the vector product \(\overrightarrow { A B } \times \overrightarrow { A C }\).
2. Hence find the equation of the plane \(A B C\) in the form \(a x + b y + c z = d\).

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

$$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + \lambda ( - 4 \mathbf { i } + \mathbf { j } ) + \mu ( 6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$

1. Find an equation of \(\Pi\) in the form \(\mathbf { r } . \mathbf { n } = p\), where \(\mathbf { n }\) is a vector perpendicular to \(\Pi\) and \(p\) is a constant. The point \(P\) has coordinates \(( 6,13,5 )\). The line \(l\) passes through \(P\) and is perpendicular to \(\Pi\). The line \(l\) intersects \(\Pi\) at the point \(N\).
2. Show that the coordinates of \(N\) are \(( 3,1 , - 1 )\). The point \(R\) lies on \(\Pi\) and has coordinates \(( 1,0,2 )\).
3. Find the perpendicular distance from \(N\) to the line \(P R\). Give your answer to 3 significant figures.
   

7. The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha \\ - 4 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right) .$$ If the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find

1. the value of \(\alpha\),
2. an equation for the plane containing the lines \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in the form \(a x + b y + c z + d = 0\), where \(a , b , c\) and \(d\) are constants. For other values of \(\alpha\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect and are skew lines.
   Given that \(\alpha = 2\),
3. find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).

5

1. Find the cartesian equation of the plane through the point ( \(2 , - 1,4\) ) with normal vector $$\mathbf { n } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right)$$
2. Find the coordinates of the point of intersection of this plane and the straight line with equation $$\mathbf { r } = \left( \begin{array} { r } 7 \\ 12 \\ 9 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right)$$

8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h] \end{figure} Relative to axes \(\mathrm { O } x\) (due east), \(\mathrm { O } y\) (due north) and \(\mathrm { O } z\) (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)

1. Verify that the cartesian equation of the plane ABC is \(3 x + 2 y + 20 z + 300 = 0\).
2. Find the vectors \(\overrightarrow { \mathrm { DE } }\) and \(\overrightarrow { \mathrm { DF } }\). Show that the vector \(2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point \(\mathrm { R } ( 15,34,0 )\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
4. Write down a vector equation of the line RS. Calculate the coordinates of S.

1 The point A has coordinates \(( 2,5,4 )\) and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8 \\ 25 \\ 43 \end{array} \right) + \lambda \left( \begin{array} { c } 4 \\ 15 \\ 25 \end{array} \right)$$ You are given that \(\mathrm { AB } = \mathrm { AC } = 15\).

1. Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
2. Find the equation of the plane ABC in cartesian form.
3. Show that the plane containing the line BC and perpendicular to the plane ABC has equation \(5 y - 3 z + 4 = 0\). The point D has coordinates \(( 1,1,3 )\).
4. Show that \(| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }\) and hence find the shortest distance between the lines \(B C\) and \(A D\).
5. Find the volume of the tetrahedron ABCD .

7 Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3 \\ ( k - 1 ) x + ( 3 - k ) y + 2 z & = 1 \\ 7 x - 3 y + 4 z & = 2 \end{aligned}$$ 7

1. The planes do not meet at a unique point.
   Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\).
   

   7

   For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)