Lines and planes in three dimensionsEdexcel International A Level Further Maths: Mind map

Exam questions on Lines and planes in three dimensions

1. The plane $\Pi$ has equation $\mathbf{r}\cdot(2\mathbf{i}-\mathbf{j}+2\mathbf{k})=6$ and the point $A$ has coordinates $(4,1,3)$.
   Write down an equation of $\Pi$ in the form $\mathbf{r}=\mathbf{a}+s\mathbf{b}+t\mathbf{c}$.2 marks
2. The line $l$ has equation $\left(\mathbf{r}-(\mathbf{i}+2\mathbf{j}-\mathbf{k})\right)\times(2\mathbf{i}-\mathbf{j}+3\mathbf{k})=\mathbf{0}$.
   Find the coordinates of the point where $l$ meets the plane $y=0$.2 marks
3. The planes $\Pi_1$ and $\Pi_2$ have equations $x+2y-z=4$ and $2x-y+3z=3$ respectively.
   Find a vector equation of the line of intersection $l$ of $\Pi_1$ and $\Pi_2$.3 marks