(1) Let A = (4, 2, -1), B = (7, 8, 8) and C = (5, 4, 2). Determine
whether or not C is on line AB. If so, find the ratio t = AC/AB and write
C as a weighted average of A and B.
(2) Let A and B be as above. Find the point D so that AD/AB = 1/2. Find
the point E so that AE/AB = -2.
(3) Let A and be as above. Write the parametric formula for line AB and
use this formula to find the point F where this line intersects the
(x,y,0) plane (i.e., the plane where the z coordinate = 0).
(4) Let A and B be as above, find the point G for which AG/GB = 3.
(5) Let A and B be as above. What is the center of mass H of a system
with mass 5 at A and mass 1 at B?
(6) Let A and B be as above, and let J be the point (0, 1, 5). Write a
parametrization of the line through J parallel to line AB. Where does
this line intersect the plane z = 0?
(7) Let A and B be as above. Then A' = (4, 2) and B' = (7, 8) are the
orthogonal projections of A and B to the plane z = 0. Suppose that K is a
point on line AB that projects to K' = (-1, -8). Check that K' is indeed
on line A'B' and then find K.
(1) Let XYZ be a triangle in the (x, y) plane. Let there be masses of 1
at X, 2 at Y, 3 at Z. What is the formula for the point P that is the
center of mass of this system. (Make up your own examples and check on
graph paper. Also, you can try X = (1, 0), Y = (0, 1), Z = (0, 0) or X =
(1, 0), Y = (2, 1), Z = (-2, -1) or any other example.)
(2) Let XYZ be a triangle in the (x, y) plane. Let there be masses of 1
at X, 2 at Y, 3 at Z. If P is the center of mass, let X' = intersection of
line XP with line YZ, Y' = intersection of line YP with line ZX, and Z' =
intersection of line ZP with line XY. Find the ratios XZ'/Z'Y, YX'/X'Z,
ZY'/ZX. Check that the relationship among the ratios stated in the Ceva
Theorem holds.
(3) Suppose that in a triangle XYZ, there are points X' on YZ and Y' on ZX
with ratios YX'/X'Z = 1/5 and ZY'/Y'X = 1/3. If P is the intersection of
lines XX' and YY', let Z' be the intersection of line ZP with line XY.
Then answer the following: (a) what is the ratio XZ'/Z'Y? (numerical
answer) (b) Find the numbers u, v, w so that P = uX + vY + wZ and u + v +
w = 1. (c) Find the numbers h and k so that h+k = 1 and Z' = hX = kY.
(4) Repeat the previous problem in the concrete case that X = (1, 0), Y = (0, 1),
Z = (0, 0). Graph it and calculate it. Check by measuring.
(5) Suppose T is an affine transformation that takes points X = (1, 0), Y
= (0, 1), Z = (0, 0) to points X' = (1,2), Y' = (23, 12) and Z' = (-1, -1).
If P = (2, 5), what is P' = T(P)?
(6) In a triangle XYZ, if X' is a point on line YZ, Y' is a point on line
ZX and Z' is a point on XY. Suppose that ratio YX'/X'Z = 1/5 and ZY'/Y'X = 1/3
and XZ'/Z'Y = 15. Are the lines XX', YY', ZZ' concurrent? If so,
what are the barycentric coordinates of the point of concurrence?
(7) Draw a triangle ABC. Let x(P), y(P), z(P) be the barycentric
coordinates with respect to this triangle, i.e. for any point P,
P = x(P)A + y(P)B + z(P) and the sum x(P)+y(P)+z(P)=1.
Now draw in the figure all points in the triangle with
(8) Draw a triangle ABC. Then
Problem 3 of
http://www.math.washington.edu/~king/coursedir/m445w01/test/quiz1/q01.html
Problem 2 of
http://www.math.washington.edu/~king/coursedir/m445w01/test/mt1/mt01.html
Problem 2 of (parts 1, 3, 4 ignore part 2)
http://www.math.washington.edu/~king/coursedir/m445w01/test/final.html
Problems E11.1, Prob 6.1, 6.4, also 6.5 is worth thinking about. (6.3 is
also an affine theorem).
http://www.math.washington.edu/~king/coursedir/m444a99/as/as06/as06.html