1.10g Problem solving with vectors: in geometry

A cyclist, when travelling due west at 15 km h\(^{-1}\), finds that the wind appears to be blowing from a bearing of 150°. When the cyclist is travelling due west at 10 km h\(^{-1}\), the wind appears to be blowing from a bearing of 135°. Find the velocity of the wind. [10]

The lines \(L_1\) and \(L_2\) have vector equations $$L_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 10 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -5 \\ 4 \end{pmatrix}$$ $$L_2 : \mathbf{r} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$$

1. Show that \(L_1\) and \(L_2\) are perpendicular. [2]
2. Show that \(L_1\) and \(L_2\) are skew lines. [3] The point \(A\) with position vector \(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) lies on \(L_2\) and the point \(X\) lies on \(L_1\) such that \(\overrightarrow{AX}\) is perpendicular to \(L_1\)
3. Find the position vector of \(X\). [5]
4. Find \(|\overrightarrow{AX}|\) [2] The point \(B\) (distinct from \(A\)) also lies on \(L_2\) and \(|\overrightarrow{BX}| = |\overrightarrow{AX}|\)
5. Find the position vector of \(B\). [5]
6. Find the cosine of angle \(AXB\). [2]