Angle with unknown parameter

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

1. In the case where \(p = 6\), find the unit vector in the direction of \(\overrightarrow { A B }\).
2. Find the values of \(p\) for which angle \(A O B = \cos ^ { - 1 } \left( \frac { 1 } { 5 } \right)\).

The equations of two straight lines are $$\mathbf{r} = \mathbf{i} + \mathbf{j} + 2a\mathbf{k} + \lambda(3\mathbf{i} + 4\mathbf{j} + a\mathbf{k}) \quad \text{and} \quad \mathbf{r} = -3\mathbf{i} - \mathbf{j} + 4\mathbf{k} + \mu(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}),$$ where \(a\) is a constant.

1. Given that the acute angle between the directions of these lines is \(\frac{1}{4}\pi\), find the possible values of \(a\). [6]
2. Given instead that the lines intersect, find the value of \(a\) and the position vector of the point of intersection. [5]

The three dimensional non-zero vector \(\mathbf{u}\) has the following properties:

• The angle \(\theta\) between \(\mathbf{u}\) and the vector \(\begin{pmatrix} 1 \\ 5 \\ 9 \end{pmatrix}\) is acute.
• The (non-reflex) angle between \(\mathbf{u}\) and the vector \(\begin{pmatrix} 9 \\ 5 \\ 1 \end{pmatrix}\) is \(2\theta\).
• \(\mathbf{u}\) is perpendicular to the vector \(\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\).

Find the angle \(\theta\). [5]