Intersections and distances in three dimensionsEdexcel A-Level Further Maths: Mind map

Exam questions on Intersections and distances in three dimensions

1. The line $l$ has equation $\mathbf{r}=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+t\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and the plane $\Pi$ has equation $2x-y+3z=14$.
   The point $A$ with position vector $(1,0,2)$ lies on $l$. Find the exact distance from $A$ to the point where $l$ meets $\Pi$.2 marks
2. The plane $\Pi$ has equation $2x-2y+z=7$ and the point $A$ has coordinates $(3,-1,4)$.
   Find the position vector of the reflection of $A$ in $\Pi$.2 marks
3. The line $l$ has equation $\mathbf{r}=\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}+\lambda\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$ and the point $P$ has coordinates $(4,2,0)$.
   Find the coordinates of the foot of the perpendicular from $P$ to $l$.3 marks