4.04b Plane equations: cartesian and vector forms

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 \(A B C\).

7 The position vectors of the points \(A , B , C , D\) are $$7 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad 11 \mathbf { i } + 3 \mathbf { j } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k } , \quad 2 \mathbf { i } + 7 \mathbf { j } + \lambda \mathbf { k }$$ respectively.

1. Given that the shortest distance between the line \(A B\) and the line \(C D\) is 3 , show that \(\lambda ^ { 2 } - 5 \lambda + 4 = 0\).
   Let \(\Pi _ { 1 }\) be the plane \(A B D\) when \(\lambda = 1\).
   Let \(\Pi _ { 2 }\) be the plane \(A B D\) when \(\lambda = 4\).
2. 1. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \mathbf { s b } + \mathbf { t c }\).
   2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(a x + b y + c z = d\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 The plane \(\Pi _ { 1 }\) has equation \(r = - 4 \mathbf { j } - 3 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( \mathbf { i } + \mathbf { j } - \mathbf { k } )\).

1. Obtain an equation of \(\Pi _ { 1 }\) in the form \(\mathrm { px } + \mathrm { qy } + \mathrm { rz } = \mathrm { d }\).
2. The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( - 5 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) = 4\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   The line \(l\) passes through the point \(A\) with position vector \(a \mathbf { i } + a \mathbf { j } + ( a - 7 ) \mathbf { k }\) and is parallel to \(( 1 - b ) \mathbf { i } + b \mathbf { j } + b \mathbf { k }\), where \(a\) and \(b\) are positive constants.
3. Given that the perpendicular distance from \(A\) to \(\Pi _ { 1 }\) is \(\sqrt { 2 }\), find the value of \(a\).
4. Given that the obtuse angle between \(l\) and \(\Pi _ { 1 }\) is \(\frac { 3 } { 4 } \pi\), find the exact value of \(b\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

6 The points \(A , B , C\) have position vectors $$\mathbf { i } + \mathbf { j } , \quad - \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , \quad - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ,$$ 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. Find a vector equation of the common perpendicular to the lines \(O C\) and \(A B\).

5 The points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , \quad 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad - 3 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$$ respectively, relative to the origin \(O\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
   The point \(D\) has position vector \(2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\).
2. Find the perpendicular distance from \(D\) to the plane \(A B C\).
3. Find the shortest distance between the lines \(A B\) and \(C D\).

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { j } - 2 \mathbf { k } )\) and \(\mathbf { r } = - 3 \mathbf { i } + 4 \mathbf { j } + \mu ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) respectively.

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
2. Obtain an equation of \(\Pi _ { 1 }\) in the form \(p x + q y + r z = s\). \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-10_2715_40_144_2007}
3. The point \(( 1,1,1 )\) lies on the plane \(\Pi _ { 2 }\). It is given that the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) passes through the point ( \(0,0,2\) ) and is parallel to the vector \(\mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\). Obtain an equation of \(\Pi _ { 2 }\) in the form \(a x + b y + c z = d\).

4 The points \(A , B , C\) have position vectors $$- \mathbf { i } + \mathbf { j } + 2 \mathbf { k } , \quad - 2 \mathbf { i } - \mathbf { j } , \quad 2 \mathbf { i } + 2 \mathbf { k } ,$$ 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. Find the acute angle between the planes \(O A B\) and \(A B C\).

7 The points \(A , B , C\) have position vectors $$- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \quad - 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } , \quad - 2 \mathbf { j } + \mathbf { k } ,$$ 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 acute angle between the planes \(O B C\) and \(A B C\).
   The point \(D\) has position vector \(t \mathbf { i } - \mathbf { j }\).
3. Given that the shortest distance between the lines \(A B\) and \(C D\) is \(\sqrt { \mathbf { 1 0 } }\), find the value of \(t\).
   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 points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } , \quad - \mathbf { j } + \mathbf { k } \quad \text { and } \quad 2 \mathbf { i } + \mathbf { j } - 7 \mathbf { k }$$ respectively, relative to the origin \(O\).

1. Find an equation of the plane \(O A B\), giving your answer in the form \(\mathbf { r } . \mathbf { n } = p\).
   The plane \(\Pi\) has equation \(\mathrm { x } - 3 \mathrm { y } - 2 \mathrm { z } = 1\).
2. Find the perpendicular distance of \(\Pi\) from the origin.
3. Find the acute angle between the planes \(O A B\) and \(\Pi\).
4. Find an equation for the common perpendicular to the lines \(O C\) and \(A B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The plane \(\Pi\) contains the lines \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\).

1. Find a Cartesian equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
   The line \(l\) passes through the point \(P\) with position vector \(2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and is parallel to the vector \(\mathbf { j } + \mathbf { k }\).
2. Find the acute angle between \(I\) and \(\Pi\).
3. Find the position vector of the foot of the perpendicular from \(P\) to \(\Pi\).

6 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 2 \mathbf { i } + \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { j } + 6 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) respectively. 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 \(l _ { 2 }\).

1. Find the length \(P Q\).
   The plane \(\Pi _ { 1 }\) contains \(P Q\) and \(l _ { 1 }\).
   The plane \(\Pi _ { 2 }\) contains \(P Q\) and \(l _ { 2 }\).
2. 1. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \mathbf { s b } + \mathbf { t c }\).
   2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(a x + b y + c z = d\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = - 2 \mathbf { i } - 3 \mathbf { j } - 5 \mathbf { k } + \lambda ( - 4 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ respectively.

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi\) contains \(l _ { 1 }\) and the point with position vector \(- \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }\).
2. Find an equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).

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 \(\Pi _ { 2 }\).

2 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } - 4 \mathbf { k } )\).
The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to the vector \(2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k }\).

1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-05_2723_33_99_22} The line \(l _ { 2 }\) is parallel to the vector \(5 \mathbf { i } - 5 \mathbf { j } - 2 \mathbf { k }\).
2. Find the acute angle between \(l _ { 2 }\) and \(\Pi\).

8 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) do not intersect and are both perpendicular to \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\). The line \(l\) intersects \(\Pi _ { 1 }\) at the point \(( 1,6,0 )\) and intersects \(\Pi _ { 2 }\) at the point \(( 3 , - 6,0 )\).

1. Find Cartesian equations of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
2. Express the vector equation of \(l\) in the form \(\left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be determined, and hence show that for points on \(l , \frac { 1 } { 2 } x + \frac { 1 } { 12 } y = 1\) and \(z = 0\). \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-16_2715_40_144_2008}
3. Show that the characteristic equation of \(\mathbf { A }\) is \(- \lambda ^ { 3 } + 3 \lambda ^ { 2 } + \frac { 7 } { 4 } \lambda = 0\) and hence find the eigenvalues of \(\mathbf { A }\). The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 3 \\ 1 & 2 & 3 \\ \frac { 1 } { 2 } & \frac { 1 } { 12 } & 0 \end{array} \right)$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-17_194_1711_484_212}
4. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }\), where \(n\) is a positive integer. [6] \includegraphics[max width=\textwidth, alt={}]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_65_1581_335_322} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_72_1579_511_324} \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_2718_35_144_2012} If you use the following page to complete the answer to any question, the question number must be clearly shown.

1. The point \(P\) has coordinates \(( 1,2,1 )\)

The line \(l\) has Cartesian equation $$\frac { x - 3 } { 5 } = \frac { y + 1 } { 3 } = \frac { z + 5 } { - 8 }$$ The plane \(\Pi _ { 1 }\) contains the point \(P\) and the line \(l\).

1. Show that a Cartesian equation for \(\Pi _ { 1 }\) is $$6 x - 2 y + 3 z = 5$$ The point \(Q\) has coordinates \(( 2 , k , - 7 )\), where \(k\) is a constant.
2. Show that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is $$\frac { 2 } { 7 } | k + 7 |$$ The plane \(\Pi _ { 2 }\) has Cartesian equation \(8 x - 4 y + z = - 3\) Given that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is the same as the shortest distance between \(\Pi _ { 2 }\) and \(Q\),
3. determine the possible values of \(k\).
   

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 plane \(\Pi\) has equation

$$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 3 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Determine a vector perpendicular to \(\Pi\) The line \(l\) meets \(\Pi\) at the point ( \(1,2,3\) ) and passes through the point ( \(1,0,1\) )
2. Determine the size of the acute angle between \(\Pi\) and \(l\) Give your answer to the nearest degree.
3. Determine the shortest distance between \(\Pi\) and the point \(( 6 , - 3 , - 6 )\)

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.

6. The coordinates of the points \(A , B\) and \(C\) relative to a fixed origin \(O\) are ( \(1,2,3\) ), \(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane \(\Pi\) contains the points \(A , B\) and \(C\).

1. Find a cartesian equation of the plane \(\Pi\). The point \(D\) has coordinates \(( k , 4,14 )\) where \(k\) is a positive constant.
   Given that the volume of the tetrahedron \(A B C D\) is 6 cubic units,
2. find the value of \(k\).
   

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9 With respect to a fixed origin \(O\), the points \(A ( - 1,5,1 ) , B ( 1,0,3 ) , C ( 2 , - 1,2 )\) and \(D ( 3,6 , - 1 )\) are the vertices of a tetrahedron.

1. Find the volume of the tetrahedron \(A B C D\). The plane \(\Pi\) contains the points \(A , B\) and \(C\).
2. Find a cartesian equation of \(\Pi\). The point \(T\) lies on the plane \(\Pi\). The line \(D T\) is perpendicular to \(\Pi\).
3. Find the exact coordinates of the point \(T\).

7. The plane \(\Pi _ { 1 }\) has equation \(x + y + z = 3\) and the plane \(\Pi _ { 2 }\) has equation \(2 x + 3 y - z = 4\) The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(L\).

1. Find a cartesian equation for the line \(L\). The plane \(\Pi _ { 3 }\) has equation $$\text { r. } \left( \begin{array} { r } 5 \\ - 4 \\ 4 \end{array} \right) = 12$$ The line \(L\) meets the plane \(\Pi _ { 3 }\) at the point \(A\).
2. Find the coordinates of \(A\).
3. Find the acute angle between \(\overrightarrow { O A }\) and the line \(L\), where \(O\) is the origin. Give your answer in degrees to one decimal place.