I'm really stuck on determining the constraints of this question in part 2.
Part one states:
Throughout this question P will be the plane in R^3 that passes
through the origin and is normal to the vector n. Furthermore, p ∈ R^3 will be a
point that is not on P and L will be the line in R^3
that passes through the point p
and is in the direction some vector v.
Draw a schematic diagram to display the above. The above I believe I can do
This is where I'm currently stuck:
Give a necessary and sufficient condition on n and v so that L intersects P
and find a formula in terms of n, p and v for this point of intersection.
I believe the above condition is that v can't be perpendicular to n. But struggle to find a formula to give the point of intersection.
Any help and links to further resources would be greatly appreciated.
You are correct that the plane and the line will intersect unless $n$ and $v$ are perpendicular. The line $L$ can be parameterised by $$L: r_0=p+\lambda v$$ with the real parameter $\lambda$. Then the plane $P$ can be written as $$P: r_1\cdot n=0$$ as the plane passes through the origin and all points on it are perpendicular to $n$ - so their dot product with $n$ is zero.
The line and plane intersect where the values of $r_0$ and $r_1$ are the same and hence $$r_0\cdot n=0$$ $$(p+\lambda v)\cdot n=0$$ for some unique value of $\lambda$ which can then give the coordinate by using $$L: r_0=p+\lambda v$$ Giving a coordinate of $p+\lambda v$ with $\lambda$ being the solution of $$(p+\lambda v)\cdot n=0$$