The lines \(L_1\) and \(L_2\) have vector equations
$$L_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 10 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -5 \\ 4 \end{pmatrix}$$
$$L_2 : \mathbf{r} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$$
- Show that \(L_1\) and \(L_2\) are perpendicular.
[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 \(L_1\)
- Find the position vector of \(X\).
[5]
- Find \(|\overrightarrow{AX}|\)
[2]
The point \(B\) (distinct from \(A\)) also lies on \(L_2\) and \(|\overrightarrow{BX}| = |\overrightarrow{AX}|\)
- Find the position vector of \(B\).
[5]
- Find the cosine of angle \(AXB\).
[2]