Is true
Within a euclidian plane R2, an isometry f is a translation if it f(x,y) = (x+a,y+b) for all points (x,y) in R2. It can also be constructed as the composition of two reflections in parallel lines.
It's not possible to answer questions like this without the diagrams and accompanying statements. This should be obvious.
I assume you mean the normal vector in the plane of the circleIf you write the circle in the form f(x,y,z) = 0 e.g. x^2 + y^2 - r^2 = 0then grad(f) gives you the normal vector (outward pointing). In cartesian (x,y,z) coordinates:grad(f) = (df/dx, df/dy, df/dz)So in our example:grad(f) = (2x, 2y, 0)This is the normal vector and is necessarily in the plane of the circle, even if this method is followed for a circle with some angle to the x-y plane :)This works for any function of the form f(...) = 0, not just circles...
If your points are (p,f), they become (p,-f).
exactly one
Is true
Is true
There are one or infinitely many points.
No. A line can be contained by many, many planes, Picture this, A rectangle with corners - going clockwise - A, B, C and D is the screen of your computer. This is a plane figure. 1 inch away from it a line runs from A1 to C1. The line is parallel to the plane. Now, take a sheet of paper with corners E, F, G and H, and place corner E at corner A of the screen, and place corner F at corner C of the screen. The Line AI is now 'contained' in the plane EFGH. and EFGH is perpendicular to ABCD.
Depends entirely on what 'F' is.
the set f all points of the plane which lie either on the circle or inside the circle form the circular region
The letter "F" is worth 4 points in Scrabble.
The fixed points of a function f(x) are the points where f(x)= x.
Both RC plane are awsome because they are both planes.
The roots of an equation of the form y = f(x), are those values of x for which y = 0. If plotted on the coordinate plane, these are the points where the graph of y against x crosses (or touches) the x axis.
Yes.
Within a euclidian plane R2, an isometry f is a translation if it f(x,y) = (x+a,y+b) for all points (x,y) in R2. It can also be constructed as the composition of two reflections in parallel lines.