Position vectors and magnitudes

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p \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 2
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- 7 \end{array} \right)$$ and angle \(A O B = 90 ^ { \circ }\).

1. Find the value of \(p\).
   The point \(C\) is such that \(\overrightarrow { O C } = \frac { 2 } { 3 } \overrightarrow { O A }\).
2. Find the unit vector in the direction of \(\overrightarrow { B C }\).
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3. Prove the identity \(\frac { 1 + \cos \theta } { \sin \theta } + \frac { \sin \theta } { 1 + \cos \theta } \equiv \frac { 2 } { \sin \theta }\).
4. 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 \(D\).

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.

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 \(11 \mathbf { i } - 8 \mathbf { j } - 6 \mathbf { k }\).

2 The points \(\mathrm { A } , \mathrm { B }\) 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 \(11 \mathbf { i } - 8 \mathbf { j } - 6 \mathbf { k }\).