[event_staff_9]

ill answer half-havocs first since there are no calculations involved; i hope im not too late for spankyou; i had just left when you posted :/

anyways, yeah the bases must be the same

correct: x^5 / x ^3 = x ^ (5-3) = x^2

incorrect: x^5 / y ^ 3 does not equal (x/y)^2

working on yours spankyou

edit: okay spankyou, to do a triple scalar product, you'll need to find a determinant of the following matrix of A, B and C:

THIS WAY MAY NOT WORK ALL THE TIME
A(1,3,2), B(3,-1,6), C(5,2,0), and D(3,6,-4)

[1 3 2
3 -1 6
5 2 0]

which is 100

note the numbers of the 3 x 3 matrix above is:

[A1 A2 A3
B1 B2 B3
C1 C2 C3]


next try, A B and D

answer: 100

lets try A, C, and D

answer: 100

how about B, C, and D now

answer: 100

so b/c you had to use triple scalar products and the question asked whether all the points lied in the same plane, and since all the answers are equal its safe to assume that they do :D

THIS WAY WORKS ALL THE TIME

heres a cooler way using a different formula:

A dot (B x C) (x = cross product) and B dot (C x D)

first you need to make three vectors. AB we'll call vector X = [ 2 -4 4] BC we'll call Y, [2 3 -6] and CD we'll call Z [-2 4 4]

so lets do X dot (Y x Z) first:

B x C = [12 20 14] dot it with A and you get 0! so A B and C are coplanar

so lets try Y dot (X x Z) X Z is [0 0 0] so the dot product must be zero! yipee! we found that A B and C lie in the same plane and B C and D also do therefore all four lie on the same plane!