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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!!!

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2 Answers 2

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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},
  {Red, Arrowheads[0.05],
   Arrow[Tube[{{0, 0, 0}, #}, 0.01]] & /@ v}},
 Axes -> True,
 BoxRatios -> {1, 1, 1}]

enter image description here

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 *)
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Use Parallelepiped:

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

Plot of parallelepiped

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

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