Determine whether there is an integer n > 1 such that there is a
projective plane of order n (i.e. with n + 1 points on each line)
such that n ̸= pk for any prime number p and integer k ≥ 1.
A projective plane is a plane (S,L ) satisfying the following
four axioms. P1. For any two distinct points P and Q there is one
and only one line containing P and Q. P2. For any two distinct
lines l and m there exists one and only one point P belonging to
l∩m. P3. There exist three noncollinear points. P4. Every line
contains at least three points.
Let π be a projective plane. Using P1 − P4, show that π...
In the real projective plane, list all possible + / - (positive
and negative) patterns for three coordinates. Then match these
triples to regions of the Fundamental Triangle. Also, describe the
location of the usual four Euclidean quadrants in the Fundamental
Triangle.
Find all primitive roots:
(a) modulo 25, or show that there are none
(b) modulo 34, or show that there are none
(c) Assuming that 2 is a primitive root modulo 67, find all
primitive roots modulo 67.
A hollow sphere is released from the top of an inclined plane of
inclination theta. (a) What should be the minimum coefficient of
friction between the plane and the sphere to prevent it from
sliding? (b) Find the kinetic energy of the sphere as it moves down
a length l on the incline if the friction coefficient is
half the value calculated in part (a).
Please show all steps
Show that the cylinder x 2 + y 2 = 4 and the sphere x 2 + y 2 +
z 2 − 8y − 6z + 21 = 0 are tangent at the point (0, 2, 3). That is,
show that the cylinder and sphere intersect at that point, and that
they share the same tangent plane at that point.
If we have two vectors v1 and v2, lie in x-y plane of the Bloch
sphere, then what the angles of phi1 and phi2 should be?
and please explain why, thanks.
Find the vector and parametric equations for the plane. The
plane that contains the lines r1(t) = <6, 8, 8,> + t<-2,
9, 6> and r2 = <6, 8, 8> + t<5, 1, 7>.