# How to get results in terms of vector operations?

Is there a simple way to simplify an expression in terms of vector operations?

For example, when I evaluate this;

v1 = {x1, y1, z1};
v2 = {x2, y2, z2};
v3 = {x3, y3, z3};
v4 = {x4, y4, z4};
Integrate[1, {x, y, z} \[Element] Tetrahedron[{v1, v2, v3, v4}]]


I get this horrible abomination;

1/6 Abs[x3 y2 z1 - x4 y2 z1 - x2 y3 z1 + x4 y3 z1 + x2 y4 z1 - x3 y4 z1 - x3 y1 z2 + x4 y1 z2 + x1 y3 z2 - x4 y3 z2 - x1 y4 z2 + x3 y4 z2 + x2 y1 z3 - x4 y1 z3 - x1 y2 z3 + x4 y2 z3 + x1 y4 z3 - x2 y4 z3 - x2 y1 z4 + x3 y1 z4 + x1 y2 z4 - x3 y2 z4 - x1 y3 z4 + x2 y3 z4]


However, this is simply the formula for the tetrahedron volume;

$$\frac16 \left| ( \vec{v}_2 - \vec{v}_1 ) \cdot ( ( \vec{v}_3 - \vec{v}_1 ) \times ( \vec{v}_4 - \vec{v}_1 ) ) \right|$$

Can Mathematica show the result I get in terms of vectors and vector operations?

There are other questions similar to this, but answers are some hacky manipualtions and not quite what I'm looking for.

Isn't there a simple, non-hacky, built-in way? There must be! C'mon Mathematica...

• You could define vectors $Assumptions = (v1 | v2 | v3 | v4) \[Element] Vectors[dim, Reals], but you need a "vectorformulated Tetrahedron"! – Ulrich Neumann Jul 1 '19 at 14:01 • I tried that. $Assumptions = (v1 | v2 | v3 | v4) \[Element] Vectors[3, Reals]; Integrate[1, {x, y, z} \[Element] Tetrahedron[{v1, v2, v3, v4}]] However, I got this error: Integrate::ilim: Invalid integration variable or limit(s) in {x,y,z}\[Element]Tetrahedron[{v1,v2,v3,v4}]. – Mahmut Akkuş Jul 1 '19 at 14:09
• Because "Tetrahedron" expects "euclidian coordinates" . You have to define your own "Tetrahedron" I think. – Ulrich Neumann Jul 1 '19 at 14:23
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• Somewhat related: mathematica.stackexchange.com/questions/3242/…, reference.wolfram.com/language/guide/SymbolicTensors.html. But you have to start with a vector/tensor formulation, perhaps. – Michael E2 Jul 1 '19 at 15:47