Intersection of lines and distancesAQA A-Level Further Maths: Mind map

Exam questions on Intersection of lines and distances

1. The line $l_1$ has equation $\mathbf r=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ and the line $l_2$ has equation $\mathbf r=\begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$.
   Show that $l_1$ and $l_2$ intersect and find the coordinates of the point of intersection.2 marks
2. The line $l_3$ has equation $\mathbf r=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ and the line $l_4$ has equation $\mathbf r=\begin{pmatrix} 1 \\ -1 \\ 5 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$.
   Explain why $l_3$ and $l_4$ are skew lines.2 marks
3. The line $l$ has equation $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$. The point $P$ has coordinates $(4,3,2)$.
   Find the coordinates of the point $F$ on $l$ for which $PF$ is perpendicular to $l$.3 marks