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 \(p\).

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 \(A B C\).

4 Relative to an origin \(O\), the position vectors of points \(P\) and \(Q\) are given by $$\overrightarrow { O P } = \left( \begin{array} { r } - 2 \\ 3 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O Q } = \left( \begin{array} { l } 2 \\ 1 \\ q \end{array} \right)$$ where \(q\) is a constant.

1. In the case where \(q = 3\), use a scalar product to show that \(\cos P O Q = \frac { 1 } { 7 }\).
2. Find the values of \(q\) for which the length of \(\overrightarrow { P Q }\) is 6 units.
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   The diagram shows the cross-section of a hollow cone and a circular cylinder. The cone has radius 6 cm and height 12 cm , and the cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The cylinder just fits inside the cone with all of its upper edge touching the surface of the cone.

7 The position vectors of the points \(A\) and \(B\), relative to an origin \(O\), are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } k \\ - k \\ 2 k \end{array} \right)$$ where \(k\) is a constant.

1. In the case where \(k = 2\), calculate angle \(A O B\).
2. Find the values of \(k\) for which \(\overrightarrow { A B }\) is a unit vector.

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.