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

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 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by $$\overrightarrow { O A } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k }$$ The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } + s ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).

1. Prove that the line \(I\) does not intersect the line through \(A\) and \(B\).
2. Find the equation of the plane containing \(l\) and the point \(A\), giving your answer in the form \(a x + b y + c z = d\).

9 The line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } + t ( 2 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )\). It is given that \(l\) lies in the plane with equation \(2 x + b y + c z = 1\), where \(b\) and \(c\) are constants.

1. Find the values of \(b\) and \(c\).
2. The point \(P\) has position vector \(2 \mathbf { j } + 4 \mathbf { k }\). Show that the perpendicular distance from \(P\) to \(l\) is \(\sqrt { } 5\).

3 Points \(A\) and \(B\) have coordinates \(( - 1,2,5 )\) and \(( 2 , - 2,11 )\) respectively. The plane \(p\) passes through \(B\) and is perpendicular to \(A B\).

1. Find an equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the acute angle between \(p\) and the \(y\)-axis.

6 The points \(P\) and \(Q\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O P } = 7 \mathbf { i } + 7 \mathbf { j } - 5 \mathbf { k } \quad \text { and } \quad \overrightarrow { O Q } = - 5 \mathbf { i } + \mathbf { j } + \mathbf { k }$$ The mid-point of \(P Q\) is the point \(A\). The plane \(\Pi\) is perpendicular to the line \(P Q\) and passes through \(A\).

1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
2. The straight line through \(P\) parallel to the \(x\)-axis meets \(\Pi\) at the point \(B\). Find the distance \(A B\), correct to 3 significant figures.

10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( a \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\), where \(a\) is a constant. The plane \(p\) has equation \(x + 2 y + 2 z = 6\). Find the value or values of \(a\) in each of the following cases.

1. The line \(l\) is parallel to the plane \(p\).
2. The line \(l\) intersects the line passing through the points with position vectors \(3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and \(\mathbf { i } + \mathbf { j } - \mathbf { k }\).
3. The acute angle between the line \(l\) and the plane \(p\) is \(\tan ^ { - 1 } 2\).

7 The straight line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k } )\). The plane \(p\) passes through the point \(( 4 , - 1,2 )\) and is perpendicular to \(l\).

1. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from the origin to \(p\).
3. A second plane \(q\) is parallel to \(p\) and the perpendicular distance between \(p\) and \(q\) is 14 units. Find the possible equations of \(q\).

10 The points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { O B } = \mathbf { i } + \mathbf { j } + 5 \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - \mathbf { k } )\).

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. Find the equation of the plane containing the line \(l\) and the point \(A\). Give your answer in the form \(a x + b y + c z = d\).

9 Two planes have equations \(x + 3 y - 2 z = 4\) and \(2 x + y + 3 z = 5\). The planes intersect in the straight line \(l\).

1. Calculate the acute angle between the two planes.
2. Find a vector equation for the line \(l\).

9 With respect to the origin \(O\), the points \(A , B , C , D\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \quad \overrightarrow { O B } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } , \quad \overrightarrow { O C } = 2 \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { O D } = - 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$$

1. Find the equation of the plane containing \(A , B\) and \(C\), giving your answer in the form \(a x + b y + c z = d\).
2. The line through \(D\) parallel to \(O A\) meets the plane with equation \(x + 2 y - z = 7\) at the point \(P\). Find the position vector of \(P\) and show that the length of \(D P\) is \(2 \sqrt { } ( 14 )\).

9 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\), \(\overrightarrow { O B } = 4 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O C } = 2 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }\). A fourth point \(D\) is such that the quadrilateral \(A B C D\) is a parallelogram.

1. Find the position vector of \(D\) and verify that the parallelogram is a rhombus.
2. The plane \(p\) is parallel to \(O A\) and the line \(B C\) lies in \(p\). Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).

8 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O B } = 2 \mathbf { i } + 3 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } - \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).

1. Show that the line passing through \(A\) and \(B\) does not intersect \(l\).
2. Show that the length of the perpendicular from \(A\) to \(l\) is \(\frac { 1 } { \sqrt { 2 } }\).

10 The points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + \mathbf { j } + \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + m \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k } )\), where \(m\) is a constant.

1. Given that the line \(l\) intersects the line passing through \(A\) and \(B\), find the value of \(m\).
2. Find the equation of the plane which is parallel to \(\mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k }\) and contains the points \(A\) and \(B\). Give your answer in the form \(a x + b y + c z = d\).

9 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } + \mathbf { j } + \mathbf { k }\) respectively. The line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. The plane \(m\) is perpendicular to \(A B\) and passes through the mid-point of \(A B\). The plane \(m\) intersects the line \(l\) at the point \(P\). Find the equation of \(m\) and the position vector of \(P\).

10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )\).

1. The point \(P\) has position vector \(4 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\). Find the length of the perpendicular from \(P\) to \(l\).
2. It is given that \(l\) lies in the plane with equation \(a x + b y + 2 z = 13\), where \(a\) and \(b\) are constants. Find the values of \(a\) and \(b\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)\). The plane \(p\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) = 6\).

1. Show that \(l\) is parallel to \(p\).
2. A line \(m\) lies in the plane \(p\) and is perpendicular to \(l\). The line \(m\) passes through the point with coordinates (5, 3, 1). Find a vector equation for \(m\).

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]

9 Two planes have equations \(2 x + 3 y - z = 1\) and \(x - 2 y + z = 3\).

1. Find the acute angle between the planes.
2. Find a vector equation for the line of intersection of the planes.

10 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = \mathbf { i } - 2 \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 6 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } + t ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$ respectively.

1. Show that \(l\) and \(m\) intersect, and find the position vector of their point of intersection.
2. Find the equation of the plane containing \(l\) and \(m\), giving your answer in the form \(a x + b y + c z = d\).

9 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = 2 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } + s ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + t ( - 2 \mathbf { i } + \mathbf { j } + \mathbf { k } )$$ respectively.

1. Show that \(l\) and \(m\) do not intersect. The point \(P\) lies on \(l\) and the point \(Q\) has position vector \(2 \mathbf { i } - \mathbf { k }\).
2. Given that the line \(P Q\) is perpendicular to \(l\), find the position vector of \(P\).
3. Verify that \(Q\) lies on \(m\) and that \(P Q\) is perpendicular to \(m\).

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 .

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

9

1. Express \(\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

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 \(l\).

7 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j }\). The point \(P\) lies on the line \(A B\) and \(O P\) is perpendicular to \(A B\).

1. Find a vector equation for the line \(A B\).
2. Find the position vector of \(P\).
3. Find the equation of the plane which contains \(A B\) and which is perpendicular to the plane \(O A B\), giving your answer in the form \(a x + b y + c z = d\).