# Drawing the volume generated by three vectors

I have three vectors:

v1={1,0,0}       v1={0,1,0}     v1=1/2{1,1,1}


I wish to show the volume constructed by these vectors.

I know the amount of this volume is calculated by

1/2  Dot[Cross[{1, 0, 0}, {0, 1, 0}], {1, 1, 1}]


But I do not know how to draw this volume!!!

To include the vectors in the drawing:

v = {{1, 0, 0}, {0, 1, 0}, 1/2 {1, 1, 1}};

ℛ = Parallelepiped[{0, 0, 0}, v];

Graphics3D[{
{Opacity[0.7], ℛ},
Arrow[Tube[{{0, 0, 0}, #}, 0.01]] & /@ v}},
Axes -> True,
BoxRatios -> {1, 1, 1}]


The volume can be calculated multiple ways

{Volume[ℛ], RegionMeasure[ℛ], Integrate[1, {x, y, z} ∈ ℛ]}

(* {1/2, 1/2, 1/2} *)

SameQ @@ %

(* True *)


Use Parallelepiped:

v1 = {1, 0, 0};
v2 = {0, 1, 0};
v3 = 1/2 {1, 1, 1};
Graphics3D[Parallelepiped[{0, 0, 0}, {v1, v2, v3}]]


• Thank you so much. But How to show this volume in a cubic with sides equal to 1 Oct 6, 2020 at 16:14
• Assuming you mean the aspect ratio, Show[Graphics3D[Parallelepiped[{0, 0, 0}, {v1, v2, v3}]], BoxRatios -> 1]. Oct 6, 2020 at 16:18