## 350 Reputation

7 years, 213 days

## Are there two vectors a and b with integ...

Maple

With two vectors a and b, we know that
Norm(CrossProduct(a, b)) = Norm(a)* Norm(b) * sin(a,b).
I tried with a := <1, 2, -2>; b := <2, 10, 11>;

Note that a perpendicular to b and

Norm(CrossProduct(a, b)) = Norm(a)* Norm(b)

I tried

restart;
with(VectorCalculus);
SetCoordinates(cartesian[x, y, z]);
a := <1, 2, -2>;
b := <2, 10, 11>;
Norm(a);
Norm(b);
v := CrossProduct(a, b);
Norm(v);

Are there two vectors a and b with integer coordinates and  not perpendicular,  so that Norm(a), Norm(b), Norm(CrossProduct(a, b)) are interger numbers satisfying
Norm(CrossProduct(a, b)) = Norm(a)* Norm(b) * sin(a,b).

## How can I find three numbers a, b, c so ...

Maple

The system of equations x*y*z + y*z + y = 21, x*y*z + x*z + z = 30, x*y*z + x*y + x = 12

has three solutions, one of them is not an integer solution.

solve({x*y*z + y*z + y = 21, x*y*z + x*z + z = 30, x*y*z + x*y + x = 12}, {x, y, z})

How can I find three numbers a, b, c so that the system of equations

x*y*z + y*z + y = a, x*y*z + x*z + z = b, x*y*z + x*y + x = c

has three  solutions (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3), where x1, y1, z1, x2, y2, z2, x3, y3, z3 are nine integer numbers.

## Tetrahedron with integer volume...

Maple

Tetrahedron with length of sides like this picture has volume is an integer number. Is there another tetrahedron like that?

## How to find six integer numbers a, b, c,...

Maple

I know that, the function f(x) = (5x^2 + 8x+ 2)/(2x^2 + 6x + 5) sastifying the conditions:

1. The solutions of the f'(x)=0 are -2 and -1;
2. f(-2) = 6 and f(-1) = -1.

How can I find six integer numbers a, b, c, d, e, m from 1 to 10 so that the function
f(x) = (a*x^2 + b*x + c)/(d*x^2 + e*x + m)
so that the equation f'(x)= 0 has two integer solutions x1, x2 and f(x1); f(x2) are also  two integer numbers?

## For what the values of integer numbers k...

Maple

I find by my hand some equations have four integer solutions.

How can I tell Maple to do this? For what the values of integer numbers k, m, n, a, b, c, d so the equation
k/(x^2 + a x  + b) +  m/(x^2 + a x  + c) + n/(x^2 + a x  + d) = 0 have four integer solutions?

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