The equations for a line and plane in vector form may be useful.

Line: ${\bf r}={\bf a} + \lambda{\bf b}$

Plane: ${\bf r}={\bf a} + \lambda{\bf b}+ \mu{\bf c}$

It may also be useful to recall that matrix multiplication is distributive:

${\bf M}({\bf a} + {\bf b}) = {\bf Ma} + {\bf Mb}$

This problem uses concepts met in the later Further Pure Maths A level modules.