I have three vectors a, b, c. Knowing that the scalar of product a . b = 1, a. c = 1, b.c=1 and modul of a equal to 1, modul of b equal to 4 and modul of c equal to 4. Now I consider two vectors u = -6*a + b + c and v = 3*a -4*b + c. How to find the angle of two vectors u and v?

I want to find the area of the triangle ABC with the sides are a, b, c. I tried

a:=sqrt(91)/6:

b:=sqrt(17)/2:

c:=sqrt(13)/3:

p:=(a+b+c)/2:

s:=simplify(sqrt(p*(p-a)*(p-b)*(p-c)));

How can I get the result sqrt(523)/24?

I want to solve the equation x^2 + y^2 = 1, where x, y are rational numbers. How can I tell Maple to do that?

How to find 2013th term in the sequence 

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...?

in which the n-th positive integer appears n times. 

I don't know how to start.

In geom3d. I want to find the vertices A(x1,y1,z1), B(x2,y2,z2), where x1, y1, z1, x2, y2, z2 are integer numbers so that the triangle OAB  (O is origin) and perimeter and area are integer numbers. I tried

> resrart:

N:=5:

L:=[]:

for x1 from -N to N do

for y1 from x1 to N do

for z1 from y1 to N do

for x2 from -N to N do

for y2 from -N to N do

for z2 from -N to N do

a:=sqrt(x1^2+y1^2+z1^2):b:=sqrt(x2^2+y2^2+z2^2):c:=sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2):

p:=(a+b+c)/2:

S:=sqrt(p*(p-a)*(p-b)*(p-c)):

if type(2*p, integer) and type(S, posint)

then L:=[op(L), [[0, 0, 0], [x1, y1, z1], [x2, y2, z2]]]: fi:

od: od: od: od: od: od:

nops(L);

But my computer runs too long. I can not receive the result. How to get the answer?

If I the length of the side are 6, 25, 29. I tried 

DirectSearch:-SolveEquations([(x2-x1)^2+(y2-y1)^2+(z2-z1)^2 = 6^2, (x3-x2)^2+(y3-y2)^2+(z3-z2)^2 = 25^2,  (x3-x1)^2+(y3-y1)^2+(z3-z1)^2 = 29^2], {abs(x1) <= 30, abs(x2) <= 20, abs(x3) <= 20, abs(y1) <= 20, abs(y2) <= 20, abs(y3) <= 20, abs(z1) <= 20,abs(z2) <= 20, abs(z3) <= 20}, assume = integer, AllSolutions, solutions = 1);