4.04a Line equations: 2D and 3D, cartesian and vector forms

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 \(a\).

7 The equations of two straight lines are $$\mathbf { r } = \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = a \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } + 3 a \mathbf { k } )$$ where \(a\) is a constant.

1. Show that the lines intersect for all values of \(a\).
2. Given that the point of intersection is at a distance of 9 units from the origin, find the possible values of \(a\).

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 \(C\).

8 Two planes have equations \(3 x + y - z = 2\) and \(x - y + 2 z = 3\).

1. Show that the planes are perpendicular.
2. Find a vector equation for the line of intersection of the two planes.

10 The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\).

1. Find the position vectors of the two points on the line whose distance from the origin is \(\sqrt { } ( 10 )\).
2. The plane \(p\) has equation \(a x + y + z = 5\), where \(a\) is a constant. The acute angle between the line \(l\) and the plane \(p\) is equal to \(\sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)\). Find the possible values of \(a\).

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

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 3 \mathbf { i } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k } )\) and \(\mathbf { r } = 3 \mathbf { i } - 5 \mathbf { j } - 6 \mathbf { k } + \mu ( 5 \mathbf { j } + 6 \mathbf { k } )\) respectively.

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to the vector \(\mathbf { i } + \mathbf { k }\).
2. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
3. Find the acute angle between \(l _ { 2 }\) and \(\Pi\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = - 5 \mathbf { j } + \lambda ( 5 \mathbf { i } + 2 \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { j } + \mathbf { k } )\) respectively. The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).

1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the distance between \(l _ { 2 }\) and \(\Pi\).
   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 \(l _ { 2 }\).
3. Show that \(P\) has position vector \(\frac { 55 } { 27 } \mathbf { i } - 5 \mathbf { j } + \frac { 22 } { 27 } \mathbf { k }\) and state a vector equation for \(P Q\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Let \(t\) be a positive constant.
The line \(l _ { 1 }\) passes through the point with position vector \(t \mathbf { i } + \mathbf { j }\) and is parallel to the vector \(- 2 \mathbf { i } - \mathbf { j }\). The line \(l _ { 2 }\) passes through the point with position vector \(\mathbf { j } + t \mathbf { k }\) and is parallel to the vector \(- 2 \mathbf { j } + \mathbf { k }\). It is given that the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\) is \(\sqrt { \mathbf { 2 1 } }\).

1. Find the value of \(t\).
   The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
2. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
   The plane \(\Pi _ { 2 }\) has Cartesian equation \(5 x - 6 y + 7 z = 0\).
3. Find the acute angle between \(l _ { 2 }\) and \(\Pi _ { 2 }\).
4. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

6 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = - \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + s ( 2 \mathbf { i } - 3 \mathbf { j } )\) and \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { k } + t ( 3 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) respectively. The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and the point \(P\) with position vector \(- 2 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }\).

1. Find an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
   The plane \(\Pi _ { 2 }\) contains \(l _ { 2 }\) and is parallel to \(l _ { 1 }\).
2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
4. The point \(Q\) is such that \(\overrightarrow { \mathrm { OQ } } = - 5 \overrightarrow { \mathrm { OP } }\). Find the position vector of the foot of the perpendicular from the point \(Q\) to \(\Pi _ { 2 }\).

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.

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 \(\Pi\).

6 The position vectors of the points \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k } ,$$ respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 .

1. Show that the only possible value of \(m\) is 2 .
2. Find the shortest distance of \(D\) from the line through \(A\) and \(C\).
3. Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right)\).

9 \includegraphics[max width=\textwidth, alt={}, center]{c1fbc9ef-2dc6-43c3-bc58-179f683c9acf-16_696_1104_264_518} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 2\) units, \(O C = 4\) units and \(O G = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O G\) respectively. The point \(M\) is the midpoint of \(D F\). The point \(N\) on \(A B\) is such that \(A N = 3 N B\).

1. Express the vectors \(\overrightarrow { O M }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Find a vector equation for the line through \(M\) and \(N\).
3. Show that the length of the perpendicular from \(O\) to the line through \(M\) and \(N\) is \(\sqrt { \frac { 53 } { 6 } }\).

9 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = - \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 5 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } + \mathbf { k } )$$ respectively, where \(a\) and \(b\) are constants.

1. Given that \(l\) and \(m\) intersect, show that \(2 b - a = 4\).
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 \(m\).

9 With respect to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow { O A } = \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\).

1. Find in degrees the acute angle between the directions of \(O A\) and \(l\).
2. Find the position vector of the foot of the perpendicular from \(A\) to \(l\).
3. Hence find the position vector of the reflection of \(A\) in \(l\).