Position vectors and magnitudes

3

1. Prove the identity \(\frac { 1 + \cos \theta } { \sin \theta } + \frac { \sin \theta } { 1 + \cos \theta } \equiv \frac { 2 } { \sin \theta }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
2. Hence solve the equation \(\frac { 1 + \cos \theta } { \sin \theta } + \frac { \sin \theta } { 1 + \cos \theta } = \frac { 3 } { \cos \theta }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 8 \\ - 6 \\ 5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 10 \\ 3 \\ - 13 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2 \\ - 3 \\ - 1 \end{array} \right)$$ A fourth point, \(D\), is such that the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\) are the first, second and third terms respectively of a geometric progression.

1. Find the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\).
2. Given that \(D\) is a point lying on the line through \(B\) and \(C\), find the two possible position vectors of the point \(D\).

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.

1. 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.
2. Find the exact value of the constant \(A\) and the exact value of the constant \(k\).
3. Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 7 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

4 You are given that \(\mathbf { a } = 2 \mathbf { i } + 6 \mathbf { j } + 9 \mathbf { k }\) and \(\mathbf { b } = \mathbf { i } + 3 \mathbf { j } - \mathbf { k }\).

1. Write down a unit vector parallel to a.
2. Find the value of \(\lambda\) such that \(\mathbf { a } + \lambda \mathbf { b }\) is parallel to \(\mathbf { k }\).
3. Calculate the size of the angle between \(\mathbf { a }\) and \(\mathbf { b }\).

2 The points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1 \\ - 2 \\ 5 \end{array} \right)\) and \(\left( \begin{array} { c } - 3 \\ - 1 \\ 2 \end{array} \right)\) respectively.

1. Find the exact length of \(A B\).
2. Find the position vector of the midpoint of \(A B\). The points \(P\) and \(Q\) have position vectors \(\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 5 \\ 1 \\ 3 \end{array} \right)\) respectively.
3. Show that \(A B P Q\) is a parallelogram.

1. Relative to a fixed origin \(O\)

• the point \(A\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\)
• the point \(B\) has position vector \(2 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k }\) where \(a\) is a positive integer.
  1. Show that \(| \overrightarrow { O A } | = \sqrt { 38 }\)
  2. Find the smallest value of \(a\) for which

$$| \overrightarrow { O B } | > | \overrightarrow { O A } |$$

3 The points \(A\) and \(B\) have position vectors \(\mathbf { a } = \left( \begin{array} { r } 3 \\ 2 \\ - 1 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } - 1 \\ 4 \\ 8 \end{array} \right)\) respectively.
Show that the exact value of the distance \(A B\) is \(\sqrt { \mathbf { 1 0 1 } }\).

1. \(\left[ \begin{array} { l } \text { With respect to the right-hand rule, a rotation through } \theta ^ { \circ } \text { anticlockwise about the } \\ y \text {-axis is represented by the matrix } \end{array} \right]\) \(\left( \begin{array} { c c c } \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ - \sin \theta & 0 & \cos \theta \end{array} \right)\)

The point \(P\) has coordinates (8, 3, 2)
The point \(Q\) is the image of \(P\) under the transformation reflection in the plane \(y = 0\)

1. Write down the coordinates of \(Q\) The point \(R\) is the image of \(P\) under the transformation rotation through \(120 ^ { \circ }\) anticlockwise about the \(y\)-axis, with respect to the right-hand rule.
2. Determine the exact coordinates of \(R\)
3. Hence find \(| \overrightarrow { P R } |\) giving your answer as a simplified surd.
4. Show that \(\overrightarrow { P R }\) and \(\overrightarrow { P Q }\) are perpendicular.
5. Hence determine the exact area of triangle \(P Q R\), giving your answer as a surd in simplest form.

1. The surface of a horizontal tennis court is modelled as part of a horizontal plane, with the origin on the ground at the centre of the court, and

• i and j are unit vectors directed across the width and length of the court respectively
• \(\quad \mathbf { k }\) is a unit vector directed vertically upwards
• units are metres

After being hit, a tennis ball, modelled as a particle, moves along the path with equation $$\mathbf { r } = \left( - 4.1 + 9 \lambda - 2.3 \lambda ^ { 2 } \right) \mathbf { i } + ( - 10.25 + 15 \lambda ) \mathbf { j } + \left( 0.84 + 0.8 \lambda - \lambda ^ { 2 } \right) \mathbf { k }$$ where \(\lambda\) is a scalar parameter with \(\lambda \geqslant 0\) Assuming that the tennis ball continues on this path until it hits the ground,

1. find the value of \(\lambda\) at the point where the ball hits the ground. The direction in which the tennis ball is moving at a general point on its path is given by $$( 9 - 4.6 \lambda ) \mathbf { i } + 15 \mathbf { j } + ( 0.8 - 2 \lambda ) \mathbf { k }$$
2. Write down the direction in which the tennis ball is moving as it hits the ground.
3. Hence find the acute angle at which the tennis ball hits the ground, giving your answer in degrees to one decimal place. The net of the tennis court lies in the plane \(\mathbf { r } . \mathbf { j } = 0\)
4. Find the position of the tennis ball at the point where it is in the same plane as the net. The maximum height above the court of the top of the net is 0.9 m .
   Modelling the top of the net as a horizontal straight line,
5. state whether the tennis ball will pass over the net according to the model, giving a reason for your answer. With reference to the model,
6. decide whether the tennis ball will actually pass over the net, giving a reason for your answer.

5 Points \(A , B\) and \(C\) have position vectors \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) , \left( \begin{array} { c } 2 \\ - 1 \\ 5 \end{array} \right)\) and \(\left( \begin{array} { c } - 4 \\ 0 \\ 3 \end{array} \right)\) respectively.

1. Find the exact distance between the midpoint of \(A B\) and the midpoint of \(B C\). Point \(D\) has position vector \(\left( \begin{array} { c } x \\ - 6 \\ z \end{array} \right)\) and the line \(C D\) is parallel to the line \(A B\).
2. Find all the possible pairs of \(x\) and \(z\).

2 The points \(A\), \(B\) and \(C\) have position vectors \(3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } , - \mathbf { i } + 6 \mathbf { k }\) and \(7 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k }\) respectively. M is the midpoint of BC .

1. Show that the magnitude of \(\overrightarrow { O M }\) is equal to \(\sqrt { 17 }\). Point D is such that \(\overrightarrow { B C } = \overrightarrow { A D }\).
2. Show that position vector of the point D is \(11 \mathbf { i } - 8 \mathbf { j } - 6 \mathbf { k }\).

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 5 \\ 4 \\ -3 \end{pmatrix}$$ The point \(P\) lies on \(AB\) and is such that \(\overrightarrow{AP} = \frac{3}{4}\overrightarrow{AB}\).

1. Find the position vector of \(P\). [3]
2. Find the distance \(OP\). [1]
3. Determine whether \(OP\) is perpendicular to \(AB\). Justify your answer. [2]