Equal length conditions

11 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$

1. Use a scalar product to find angle \(A O B\), correct to the nearest degree.
2. Find the unit vector in the direction of \(\overrightarrow { A B }\).
3. The point \(C\) is such that \(\overrightarrow { O C } = 6 \mathbf { j } + p \mathbf { k }\), where \(p\) is a constant. Given that the lengths of \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\) are equal, find the possible values of \(p\). \footnotetext{Every reasonable effort has been made to trace all copyright holders where the publishers (i.e. UCLES) are aware that third-party material has been reproduced. The publishers would be pleased to hear from anyone whose rights they have unwittingly infringed.
   University of Cambridge International Examinations is part of the University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

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2 \end{array} \right)$$

1. Find the cosine of angle \(A O B\). The position vector of \(C\) is given by \(\overrightarrow { O C } = \left( \begin{array} { c } k \\ - 2 k \\ 2 k - 3 \end{array} \right)\).
2. Given that \(A B\) and \(O C\) have the same length, find the possible values of \(k\). 5 A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius \(r \mathrm {~cm}\).
3. Show that the area of the sector, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 12 r - r ^ { 2 }\).
4. Express \(A\) in the form \(a - ( r - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
5. Given that \(r\) can vary, state the greatest value of \(A\) and find the corresponding angle of the sector. [2] 6 The line with gradient - 2 passing through the point \(P ( 3 t , 2 t )\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
6. Find the area of triangle \(A O B\) in terms of \(t\). The line through \(P\) perpendicular to \(A B\) intersects the \(x\)-axis at \(C\).
7. Show that the mid-point of \(P C\) lies on the line \(y = x\).

9 The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { c } 1 \\ 5 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right) ,$$ where \(p\) is a constant.

1. Find the value of \(p\) for which the lengths of \(A B\) and \(C B\) are equal.
2. For the case where \(p = 1\), use a scalar product to find angle \(A B C\).

2. Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) the point \(B\) has position vector ( \(\mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }\) )
and the point \(C\) has position vector \(( - \mathbf { i } + \mathbf { j } + a \mathbf { k } )\), where \(a\) is a constant and \(a > 0\).
Given that \(| \overrightarrow { B C } | = \sqrt { 41 }\) a. show that \(a = 2\). \(D\) is the point such that \(A B C D\) forms a parallelogram.
b. Find the position vector of \(D\).

10 Point A has position vector \(\left( \begin{array} { l } a \\ b \\ 0 \end{array} \right)\) where \(a\) and \(b\) can vary, point B has position vector \(\left( \begin{array} { l } 4 \\ 2 \\ 0 \end{array} \right)\) and point C has position vector \(\left( \begin{array} { l } 2 \\ 4 \\ 2 \end{array} \right)\). ABC is an isosceles triangle with \(\mathrm { AC } = \mathrm { AB }\).

1. Show that \(a - b + 1 = 0\).
2. Determine the position vector of A such that triangle ABC has minimum area. Answer all the questions.
   Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.