1.10f Distance between points: using position vectors

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

1. Find a vector equation for the line through \(A\) and \(B\).
2. Find the acute angle between the directions of \(A B\) and \(l\), giving your answer in degrees.
3. Show that the line through \(A\) and \(B\) does not intersect the line \(l\).

10 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + 7 x + 8 } { ( 1 + x ) ( 2 + x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{98001cfe-46a1-4c8f-9230-c140ebff6176-18_737_1034_262_552} In the diagram, \(O A B C D\) is a solid figure in which \(O A = O B = 4\) units and \(O D = 3\) units. The edge \(O D\) is vertical, \(D C\) is parallel to \(O B\) and \(D C = 1\) unit. The base, \(O A B\), is horizontal and angle \(A O B = 90 ^ { \circ }\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O D\) respectively. The midpoint of \(A B\) is \(M\) and the point \(N\) on \(B C\) is such that \(C N = 2 N B\).
   1. Express vectors \(\overrightarrow { M D }\) and \(\overrightarrow { O N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
   2. Calculate the angle in degrees between the directions of \(\overrightarrow { M D }\) and \(\overrightarrow { O N }\).
   3. Show that the length of the perpendicular from \(M\) to \(O N\) is \(\sqrt { \frac { 22 } { 5 } }\).
      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 Relative to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 3 \\ 1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 5 \\ 3 \\ - 2 \end{array} \right)$$

1. Using a scalar product, find the cosine of angle \(B A C\).
2. Hence find the area of triangle \(A B C\). Give your answer in a simplified exact form.

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} The curve \(C _ { 1 }\) has equation $$x y = \frac { 15 } { 2 } - 5 x \quad x \neq 0$$ The curve \(C _ { 2 }\) has equation $$y = x ^ { 3 } - \frac { 7 } { 2 } x - 5$$

1. Show that \(C _ { 1 }\) and \(C _ { 2 }\) meet when $$2 x ^ { 4 } - 7 x ^ { 2 } - 15 = 0$$ Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at points \(P\) and \(Q\)
2. find, using algebra, the exact distance \(P Q\)

1. The line \(l _ { 1 }\) has equation \(x + 3 y - 11 = 0\)

The point \(A\) and the point \(B\) lie on \(l _ { 1 }\) Given that \(A\) has coordinates ( \(- 1 , p\) ) and \(B\) has coordinates ( \(q , 2\) ), where \(p\) and \(q\) are integers,

1. find the value of \(p\) and the value of \(q\),
2. find the length of \(A B\), giving your answer as a simplified surd. The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the midpoint of \(A B\).
3. Find an equation for \(l _ { 2 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.

7. The point \(A\) has coordinates \(( - 1,5 )\) and the point \(B\) has coordinates \(( 4,1 )\). The line \(l\) passes through the points \(A\) and \(B\).

1. Find the gradient of \(l\).
2. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. The point \(M\) is the midpoint of \(A B\). The point \(C\) has coordinates \(( 5 , k )\) where \(k\) is a constant.
   Given that the distance from \(M\) to \(C\) is \(\sqrt { 13 }\)
3. find the exact possible values of the constant \(k\).
   

4. The point \(A ( - 6,4 )\) and the point \(B ( 8 , - 3 )\) lie on the line \(L\).

1. Find an equation for \(L\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
2. Find the distance \(A B\), giving your answer in the form \(k \sqrt { 5 }\), where \(k\) is an integer.

1. The curve \(C\) has equation

$$y = 9 - 4 x - \frac { 8 } { x } , \quad x > 0$$ The point \(P\) on \(C\) has \(x\)-coordinate equal to 2 .

1. Show that the equation of the tangent to \(C\) at the point \(P\) is \(y = 1 - 2 x\).
2. Find an equation of the normal to \(C\) at the point \(P\). The tangent at \(P\) meets the \(x\)-axis at \(A\) and the normal at \(P\) meets the \(x\)-axis at \(B\).
3. Find the area of triangle \(A P B\).

1. (a) Factorise completely \(x ^ { 3 } - 4 x\) (b) Sketch the curve \(C\) with equation

$$y = x ^ { 3 } - 4 x ,$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. The point \(A\) with \(x\)-coordinate - 1 and the point \(B\) with \(x\)-coordinate 3 lie on the curve \(C\).
(c) Find an equation of the line which passes through \(A\) and \(B\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
(d) Show that the length of \(A B\) is \(k \sqrt { } 10\), where \(k\) is a constant to be found.

9. The line \(L _ { 1 }\) has equation \(2 y - 3 x - k = 0\), where \(k\) is a constant. Given that the point \(A ( 1,4 )\) lies on \(L _ { 1 }\), find

1. the value of \(k\),
2. the gradient of \(L _ { 1 }\). The line \(L _ { 2 }\) passes through \(A\) and is perpendicular to \(L _ { 1 }\).
3. Find an equation of \(L _ { 2 }\) giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The line \(L _ { 2 }\) crosses the \(x\)-axis at the point \(B\).
4. Find the coordinates of \(B\).
5. Find the exact length of \(A B\).

10. The points \(Q ( 1,3 )\) and \(R ( 7,0 )\) lie on the line \(l _ { 1 }\), as shown in Figure 2.
The length of \(Q R\) is \(a \sqrt { } 5\).

1. Find the value of \(a\). The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\), passes through \(Q\) and crosses the \(y\)-axis at the point \(P\), as shown in Figure 2. Find
2. an equation for \(l _ { 2 }\),
3. the coordinates of \(P\),
4. the area of \(\triangle P Q R\).

8. \begin{figure}[h] \end{figure} The points \(A\) and \(B\) have coordinates \(( 6,7 )\) and \(( 8,2 )\) respectively.
The line \(l\) passes through the point \(A\) and is perpendicular to the line \(A B\), as shown in Figure 1.

1. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. Given that \(l\) intersects the \(y\)-axis at the point \(C\), find
2. the coordinates of \(C\),
3. the area of \(\triangle O C B\), where \(O\) is the origin.

8. (a) Find an equation of the line joining \(A ( 7,4 )\) and \(B ( 2,0 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
(b) Find the length of \(A B\), leaving your answer in surd form. The point \(C\) has coordinates ( \(2 , t\) ), where \(t > 0\), and \(A C = A B\).
(c) Find the value of \(t\).
(d) Find the area of triangle \(A B C\). \(\_\_\_\_\)

11. \begin{figure}[h] \end{figure} The line \(y = x + 2\) meets the curve \(x ^ { 2 } + 4 y ^ { 2 } - 2 x = 35\) at the points \(A\) and \(B\) as shown in Figure 2.

1. Find the coordinates of \(A\) and the coordinates of \(B\).
2. Find the distance \(A B\) in the form \(r \sqrt { 2 }\) where \(r\) is a rational number.

1. (a) Factorise completely \(9 x - 4 x ^ { 3 }\) (b) Sketch the curve \(C\) with equation

$$y = 9 x - 4 x ^ { 3 }$$ Show on your sketch the coordinates at which the curve meets the \(x\)-axis. The points \(A\) and \(B\) lie on \(C\) and have \(x\) coordinates of - 2 and 1 respectively.
(c) Show that the length of \(A B\) is \(k \sqrt { } 10\) where \(k\) is a constant to be found.

5. $$f ( x ) = x ^ { 2 } - 8 x + 19$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) crosses the \(y\)-axis at the point \(P\) and has a minimum point at the point \(Q\).
2. Sketch the graph of \(C\) showing the coordinates of point \(P\) and the coordinates of point \(Q\).
3. Find the distance \(P Q\), writing your answer as a simplified surd.

11. With respect to a fixed origin \(O\) the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 14 \\ - 6 \\ - 13 \end{array} \right) + \lambda \left( \begin{array} { r } - 2 \\ 1 \\ 4 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } p \\ - 7 \\ 4 \end{array} \right) + \mu \left( \begin{array} { l } q \\ 2 \\ 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) and \(q\) are constants. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,

1. show that \(q = 3\) Given further that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at point \(X\), find
2. the value of \(p\),
3. the coordinates of \(X\). The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { r } 6 \\ - 2 \\ 3 \end{array} \right)\) Given that point \(B\) also lies on \(l _ { 1 }\) and that \(A B = 2 A X\)
4. find the two possible position vectors of \(B\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 11 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

1. \(A B C D\) is a parallelogram with \(A B\) parallel to \(D C\) and \(A D\) parallel to \(B C\). The position vectors of \(A , B , C\), and \(D\) relative to a fixed origin \(O\) are \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) respectively.

Given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + 6 \mathbf { k } , \quad \mathbf { c } = - \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }$$

1. find the position vector \(\mathbf { d }\),
2. find the angle between the sides \(A B\) and \(B C\) of the parallelogram,
3. find the area of the parallelogram \(A B C D\). The point \(E\) lies on the line through the points \(C\) and \(D\), so that \(D\) is the midpoint of \(C E\).
4. Use your answer to part (c) to find the area of the trapezium \(A B C E\).

4. \begin{figure}[h] \end{figure} Figure 2 shows the points \(A\) and \(B\) with position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, relative to a fixed origin \(O\). Given that \(| \mathbf { a } | = 5 , | \mathbf { b } | = 6\) and a.b \(= 20\)

1. find the cosine of angle \(A O B\),
2. find the exact length of \(A B\).
3. Show that the area of triangle \(O A B\) is \(5 \sqrt { 5 }\)

8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 6 \\ 4 \\ 1 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(P\) is on \(l _ { 1 }\) where \(\lambda = 0\), and the point \(Q\) is on \(l _ { 2 }\) where \(\mu = - 1\)
2. Find the acute angle between the line segment \(P Q\) and \(l _ { 1 }\), giving your answer in degrees to 2 decimal places.
3. Find the shortest distance from the point \(Q\) to the line \(l _ { 1 }\), giving your answer to 3 significant figures.

6. The line \(l _ { 1 }\) has vector equation $$\mathbf { r } = 8 \mathbf { i } + 12 \mathbf { j } + 14 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) ,$$ where \(\lambda\) is a parameter. The point \(A\) has coordinates (4, 8, a), where \(a\) is a constant. The point \(B\) has coordinates ( \(b , 13,13\) ), where \(b\) is a constant. Points \(A\) and \(B\) lie on the line \(l _ { 1 }\).

1. Find the values of \(a\) and \(b\). Given that the point \(O\) is the origin, and that the point \(P\) lies on \(l _ { 1 }\) such that \(O P\) is perpendicular to \(l _ { 1 }\),
2. find the coordinates of \(P\).
3. Hence find the distance \(O P\), giving your answer as a simplified surd.

6. The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\) respectively. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l _ { 1 }\). A second line \(l _ { 2 }\) passes through the origin and is parallel to the vector \(\mathbf { i } + \mathbf { k }\). The line \(l _ { 1 }\) meets the line \(l _ { 2 }\) at the point \(C\).
3. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
4. Find the position vector of the point \(C\).

7. Relative to a fixed origin \(O\), the point \(A\) has position vector ( \(2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }\) ), the point \(B\) has position vector \(( 5 \mathbf { i } + 2 \mathbf { j } + 10 \mathbf { k } )\), and the point \(D\) has position vector \(( - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\). The line \(l\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l\).
3. Show that the size of the angle \(B A D\) is \(109 ^ { \circ }\), to the nearest degree. The points \(A , B\) and \(D\), together with a point \(C\), are the vertices of the parallelogram \(A B C D\), where \(\overrightarrow { A B } = \overrightarrow { D C }\).
4. Find the position vector of \(C\).
5. Find the area of the parallelogram \(A B C D\), giving your answer to 3 significant figures.
6. Find the shortest distance from the point \(D\) to the line \(l\), giving your answer to 3 significant figures.