Find the equation of the plane normal to when .
Sorry not sure how to display pi.
Where do I start?
r(1) gives the position vector of a point in the plane.
So you have a normal to the plane and a point in the plane.
The equation of a plane is ax + by + cz = d where <a, b, c> is a normal to the plane and d is found by substituting a known point into ax + by + cz = d (once you've substituted the values of a, b and c of course).
I'm assuming that is meant to represent the x,y and z co-ordinates (respectively).
If that is the case then, when t=1, we have .
Recall the dot product - , where is the angle between the two vectors.
Note that, if two vectors are orthogonal then and so (as the cosine of is zero).
Hence, you want to find another vector, call this and find .
In other, words, you are looking for solutions to the equation . This implies that and hence there are infinitely many solutions of this form
Also, as a sidenote, if t is not constant, you get a really interesting kind of geometry based on spirals. It 'looks' a little bit like making your way from the top of a trumpet to the bottom (or vice versa, depending) only going around the inside of the horn (I think that's what it's called)
Hope that helps