Area calculations using vectors

6 Relative to an origin \(O\), the position vector of \(A\) is \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and the position vector of \(B\) is \(7 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\).

1. Show that angle \(O A B\) is a right angle.
2. Find the area of triangle \(O A B\).

9 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-16_533_601_258_772} The diagram shows a trapezium \(O A B C\) in which \(O A\) is parallel to \(C B\). The position vectors of \(A\) and \(B\) relative to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 2 \\ - 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { l } 6 \\ 1 \\ 1 \end{array} \right)\).

1. Show that angle \(O A B\) is \(90 ^ { \circ }\).
   The magnitude of \(\overrightarrow { C B }\) is three times the magnitude of \(\overrightarrow { O A }\).
2. Find the position vector of \(C\).
3. Find the exact area of the trapezium \(O A B C\), giving your answer in the form \(a \sqrt { } b\), where \(a\) and \(b\) are integers.

10 \includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-16_318_1006_260_568} Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\), shown in the diagram, are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 1 \\ 3 \\ - 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 4 \\ - 2 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 2 \\ 2 \\ - 1 \end{array} \right) .$$

1. Show that \(A B\) is perpendicular to \(B C\).
2. Show that \(A B C D\) is a trapezium.
3. Find the area of \(A B C D\), giving your answer correct to 2 decimal places.

11 \end{array} \right) \quad \text { and } \quad \overrightarrow { O X } = \left( \begin{array} { r } - 2
- 2
5 \end{array} \right)$$

1. Find \(\overrightarrow { A X }\) and show that \(A X B\) is a straight line.
   The position vector of a point \(C\) is given by \(\overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 8 \\ 3 \end{array} \right)\).
2. Show that \(C X\) is perpendicular to \(A X\).
3. Find the area of triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-18_949_1087_260_529} The diagram shows part of the curve \(y = ( x - 1 ) ^ { - 2 } + 2\), and the lines \(x = 1\) and \(x = 3\). The point \(A\) on the curve has coordinates \(( 2,3 )\). The normal to the curve at \(A\) crosses the line \(x = 1\) at \(B\).
4. Show that the normal \(A B\) has equation \(y = \frac { 1 } { 2 } x + 2\).
5. Find, showing all necessary working, the volume of revolution obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

14. Relative to a fixed origin \(O\), the line \(l\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 4 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. Points \(A\) and \(B\) lie on the line \(l\), where \(A\) has coordinates ( \(1 , a , 5\) ) and \(B\) has coordinates ( \(b , - 1,3\) ).

1. Find the value of the constant \(a\) and the value of the constant \(b\).
2. Find the vector \(\overrightarrow { A B }\). The point \(C\) has coordinates ( \(4 , - 3,2\) )
3. Show that the size of the angle \(C A B\) is \(30 ^ { \circ }\)
4. Find the exact area of the triangle \(C A B\), giving your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined. The point \(D\) lies on the line \(l\) so that the area of the triangle \(C A D\) is twice the area of the triangle \(C A B\).
5. Find the coordinates of the two possible positions of \(D\).

1. Relative to a fixed origin \(O\),
   the point \(A\) has position vector \(( 2 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\) the point \(B\) has position vector \(( 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )\) the point \(C\) has position vector ( \(2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\) )

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 \(C A B\) is \(62.8 ^ { \circ }\), to one decimal place.
4. Hence find the area of triangle \(C A B\), giving your answer to 3 significant figures. The point \(D\) lies on the line \(l\). Given that the area of triangle \(C A D\) is twice the area of triangle \(C A B\),
5. find the two possible position vectors of point \(D\).
   

1. Relative to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations

$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 2 \\ 1 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 7 \end{array} \right) + \mu \left( \begin{array} { l } 8 \\ 4 \\ 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\).

1. Write down the coordinates of \(A\). Given that the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\),
2. show that \(\sin \theta = k \sqrt { 2 }\), where \(k\) is a rational number to be found. The point \(B\) lies on \(l _ { 1 }\) where \(\lambda = 4\) The point \(C\) lies on \(l _ { 2 }\) such that \(A C = 2 A B\).
3. Find the exact area of triangle \(A B C\).
4. Find the coordinates of the two possible positions of \(C\).
   

11. Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 4 \\ 3 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) passes through the origin and is parallel to \(l _ { 1 }\)

1. Find a vector equation for \(l _ { 2 }\) The point \(A\) and the point \(B\) both lie on \(l _ { 1 }\) with parameters \(\lambda = 0\) and \(\lambda = 3\) respectively.
   Write down
2. 1. the coordinates of \(A\),
   2. the coordinates of \(B\).
3. Find the size of the acute angle between \(O A\) and \(l _ { 1 }\) Give your answer in degrees to one decimal place. The point \(D\) lies on \(l _ { 2 }\) such that \(O A B D\) is a parallelogram.
4. Find the area of \(O A B D\), giving your answer to the nearest whole number.

1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\), where \(\lambda\) is a scalar parameter.

The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 0 \\ 9 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right)\), where \(\mu\) is a scalar parameter.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(C\), find

1. the coordinates of \(C\). The point \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 0\) and the point \(B\) is the point on \(l _ { 2 }\) where \(\mu = - 1\)
2. Find the size of the angle \(A C B\). Give your answer in degrees to 2 decimal places.
3. Hence, or otherwise, find the area of the triangle \(A B C\).

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.

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

$$\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 1 \\ 4 \end{array} \right)$$ and the line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } 0 \\ 4 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right) ,$$ where \(\lambda\) and \(\mu\) are parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(B\) and the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\).

1. Find the coordinates of \(B\).
2. Find the value of \(\cos \theta\), giving your answer as a simplified fraction. The point \(A\), which lies on \(l _ { 1 }\), has position vector \(\mathbf { a } = 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\).
   The point \(C\), which lies on \(l _ { 2 }\), has position vector \(\mathbf { c } = 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k }\).
   The point \(D\) is such that \(A B C D\) is a parallelogram.
3. Show that \(| \overrightarrow { A B } | = | \overrightarrow { B C } |\).
4. Find the position vector of the point \(D\).

7. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\), where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 0 \\ 9 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right)\), where \(\mu\) is a scalar parameter.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(C\), find

1. the coordinates of \(C\). The point \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 0\) and the point \(B\) is the point on \(l _ { 2 }\) where \(\mu = - 1\).
2. Find the size of the angle \(A C B\). Give your answer in degrees to 2 decimal places.
3. Hence, or otherwise, find the area of the triangle \(A B C\).

7
0
7 \end{array} \right) + \mu \left( \begin{array} { r } 3
- 5
4 \end{array} \right) , \quad \text { where } \mu \text { is a scalar parameter. }$$

1. Find the value of the constant \(p\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the position vector of their point of intersection, \(C\).
3. Find the size of the angle \(A C B\), giving your answer in degrees to 3 significant figures.
4. Find the area of the triangle \(A B C\), giving your answer to 3 significant figures.\\ 7. The rate of increase of the number, \(N\), of fish in a lake is modelled by the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { ( k t - 1 ) ( 5000 - N ) } { t } \quad t > 0 , \quad 0 < N < 5000$$ In the given equation, the time \(t\) is measured in years from the start of January 2000 and \(k\) is a positive constant.
5. By solving the differential equation, show that $$N = 5000 - A t \mathrm { e } ^ { - k t }$$ where \(A\) is a positive constant. After one year, at the start of January 2001, there are 1200 fish in the lake. After two years, at the start of January 2002, there are 1800 fish in the lake.
6. Find the exact value of the constant \(A\) and the exact value of the constant \(k\).
7. Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish.

8. Relative to a fixed origin \(O\), the point \(A\) has position vector \(\left( \begin{array} { r } - 2 \\ 4 \\ 7 \end{array} \right)\) and the point \(B\) has position vector \(\left( \begin{array} { r } - 1 \\ 3 \\ 8 \end{array} \right)\) The line \(l _ { 1 }\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Hence find a vector equation for the line \(l _ { 1 }\) The point \(P\) has position vector \(\left( \begin{array} { l } 0 \\ 2 \\ 3 \end{array} \right)\) Given that angle \(P B A\) is \(\theta\),
3. show that \(\cos \theta = \frac { 1 } { 3 }\) The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
4. Find a vector equation for the line \(l _ { 2 }\) The points \(C\) and \(D\) both lie on the line \(l _ { 2 }\) Given that \(A B = P C = D P\) and the \(x\) coordinate of \(C\) is positive,
5. find the coordinates of \(C\) and the coordinates of \(D\).
6. find the exact area of the trapezium \(A B C D\), giving your answer as a simplified surd.

7. The point \(A\) with coordinates ( \(- 3,7,2\) ) lies on a line \(l _ { 1 }\) The point \(B\) also lies on the line \(l _ { 1 }\) Given that \(\quad \overrightarrow { A B } = \left( \begin{array} { r } 4 \\ - 6 \\ 2 \end{array} \right)\),

1. find the coordinates of point \(B\). The point \(P\) has coordinates ( \(9,1,8\) )
2. Find the cosine of the angle \(P A B\), giving your answer as a simplified surd.
3. Find the exact area of triangle \(P A B\), giving your answer in its simplest form. The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
4. Find a vector equation for the line \(l _ { 2 }\) The point \(Q\) lies on the line \(l _ { 2 }\) Given that the line segment \(A P\) is perpendicular to the line segment \(B Q\),
5. find the coordinates of the point \(Q\).

4. Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { c } 6 \\ 3 \\ - 6 \end{array} \right)\) respectively. Find, in exact, simplified form,

1. the cosine of \(\angle A O B\),
2. the area of triangle \(O A B\),
3. the shortest distance from \(A\) to the line \(O B\).

6. Relative to a fixed origin, the points \(A , B\) and \(C\) have position vectors ( \(2 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }\) ), \(( 5 \mathbf { i } - 4 \mathbf { j } )\) and \(( 7 \mathbf { i } - 6 \mathbf { j } - 4 \mathbf { k } )\) respectively.

1. Show that \(A , B\) and \(C\) all lie on a single straight line.
2. Write down the ratio \(A B : B C\) The point \(D\) has position vector \(( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\).
3. Show that \(A D\) is perpendicular to \(B D\).
4. Find the exact area of triangle \(A B D\).

8. The points \(A\) and \(B\) have coordinates \(( 3,9 , - 7 )\) and \(( 13 , - 6 , - 2 )\) respectively.

1. Find, in vector form, an equation for the line \(l\) which passes through \(A\) and \(B\).
2. Show that the point \(C\) with coordinates \(( 9,0 , - 4 )\) lies on \(l\). The point \(D\) is the point on \(l\) closest to the origin, \(O\).
3. Find the coordinates of \(D\).
4. Find the area of triangle \(O A B\) to 3 significant figures.

5 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

3 A triangle ABC has vertices \(\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )\) and \(\mathrm { C } ( 4,8,3 )\). By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle. [6]

11. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 7 \\ 4 \\ 9 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 6 \\ - 7 \\ 3 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 4 \\ b \end{array} \right) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(b\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(X\),

1. show that \(b = - 3\) and find the coordinates of \(X\). The point \(A\) lies on \(l _ { 1 }\) and has coordinates (6, 3, 5)
   The point \(B\) lies on \(l _ { 2 }\) and has coordinates \(( 14,9 , - 9 )\)
2. Show that angle \(A X B = \arccos \left( - \frac { 1 } { 10 } \right)\)
3. Using the result obtained in part (b), find the exact area of triangle \(A X B\). Write your answer in the form \(p \sqrt { q }\) where \(p\) and \(q\) are integers to be determined.

7 Points \(A ( 2,2,5 ) , B ( 1 , - 1 , - 4 ) , C ( 3,3,10 )\) and \(D ( 8,6,3 )\) are the vertices of a pyramid with a triangular base.

1. Calculate the lengths \(A B\) and \(A C\), and the angle \(B A C\).
2. Show that \(\overrightarrow { A D }\) is perpendicular to both \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\).
3. Calculate the volume of the pyramid \(A B C D\).
   [0pt] [The volume of the pyramid is \(V = \frac { 1 } { 3 } \times\) base area × perpendicular height.]

2 A triangle has vertices at \(A ( 1,1,3 ) , B ( 5,9 , - 5 )\) and \(C ( 6,5 , - 4 ) . P\) is the point on \(A B\) such that \(A P : P B = 3 : 1\).

1. Show that \(\overrightarrow { C P }\) is perpendicular to \(\overrightarrow { A B }\).
2. Find the area of the triangle \(A B C\).

4 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

9. The equations of the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) , \\ l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) , \end{array}$$ where \(\lambda\) and \(\mu\) are parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of \(Q\), their point of intersection.
2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\). The point \(P\) with \(x\)-coordinate 3 lies on the line \(l _ { 1 }\) and the point \(R\) with \(x\)-coordinate 4 lies on the line \(l _ { 2 }\).
3. Find, in its simplest form, the exact area of the triangle \(P Q R\). END