Vector and Cartesian equations of a lineEdexcel A-Level Further Maths: Mind map

Exam questions on Vector and Cartesian equations of a line

1. The line $l_1$ passes through the points $A(2,-1,4)$ and $B(5,1,0)$.
   Determine whether the point $C(11,5,-12)$ lies on $l_1$.2 marks
2. The lines $l_2$ and $l_3$ have vector equations $\mathbf r=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $\mathbf r=\begin{pmatrix} 5 \\ 0 \\ 4 \end{pmatrix}+t\begin{pmatrix} -4 \\ 2 \\ -2 \end{pmatrix}$.
   Explain why $l_2$ and $l_3$ do not intersect.2 marks
3. Three lines are given by $l_4$: $\mathbf r=\begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$, $l_5$: $\mathbf r=\begin{pmatrix} 3 \\ 6 \\ -7 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $l_6$: $\mathbf r=\begin{pmatrix} 1 \\ 4 \\ 0 \end{pmatrix}+\nu\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$.
   Show that $l_4$ and $l_5$ intersect and find the position vector of their point of intersection.3 marks