Exercise: A parallelepiped is a six-sided prism whose sides are parallelograms.
  

 
a) Show that w  u  v  gives the volume of the parallelepiped spanned by u , v , and w .

 
Let 𝑉 be the volume of the parallelepiped spanned by u , v , and w .
𝑉 = (area of the base)(height)

w

v
The base of the solid is a parallelogram.
From an example in class, we know that the area of the
parallelogram spanned by 𝐮
⃗ and 𝐯⃗ is ‖𝐮
⃗ × 𝐯⃗‖.

u
Thus, area of the base = ‖𝐮
⃗ × 𝐯⃗‖.
The height (or altitude) of the parallelepiped is the magnitude of the projection of 𝐰
⃗⃗ onto a
vector perpendicular to the base. In other words, the height is the projection of 𝐰
⃗⃗ onto 𝐮
⃗ × 𝐯⃗.
height = ‖proj𝐮⃗× 𝐯⃗ (𝐰
⃗⃗ )‖ = ‖
=|
=
(𝐮
⃗ × 𝐯⃗) ⋅ 𝐰
⃗⃗
(𝐮
⃗ × 𝐯⃗)‖
(𝐮
⃗ × 𝐯⃗) ⋅ (𝐮
⃗ × 𝐯⃗)
(𝐮
|(𝐮
⃗ × 𝐯⃗) ⋅ 𝐰
⃗⃗
⃗ × 𝐯⃗) ⋅ 𝐰
⃗⃗ |
‖𝐮
| ‖𝐮
⃗ × 𝐯⃗‖ =
⃗ × 𝐯⃗‖
(𝐮
|(𝐮
⃗ × 𝐯⃗) ⋅ (𝐮
⃗ × 𝐯⃗)
⃗ × 𝐯⃗) ⋅ (𝐮
⃗ × 𝐯⃗)|
|(𝐮
|(𝐮
⃗ × 𝐯⃗) ⋅ 𝐰
⃗⃗ |
⃗ × 𝐯⃗) ⋅ 𝐰
⃗⃗ |
‖𝐮
⃗ × 𝐯⃗‖ =
𝟐
‖𝐮
‖𝐮
⃗ × 𝐯⃗‖
⃗ × 𝐯⃗‖
𝑉 = (area of the base)(height) = ‖𝐮
⃗ × 𝐯
⃗‖
|(𝐮
⃗ × 𝐯
⃗)⋅𝐰
⃗⃗ |
= |(𝐮
⃗ × 𝐯
⃗)⋅𝐰
⃗⃗ |
‖𝐮
⃗ × 𝐯
⃗‖
= |𝐰
⃗⃗ ⋅ (𝐮
⃗ × 𝐯⃗)|
b) Determine the volume of the parallelepiped determined by the vectors [2, 3, 1],
[1, −1, 2] and [−1, 4, −4].
Volume = |([2, 3, 1] × [1, −1, 2]) ⋅ [−1, 4, −4]|
𝐢̂
[2, 3, 1] × [1, −1, 2] = |2
1
𝐣̂
3
−1
̂
𝐤
3
1| = 𝐢̂ |−1
2
1
2
| − 𝐣̂ |
2
1
̂ = [7, −3, −5]
= 7𝐢̂ − 3𝐣̂ − 5𝐤
1
̂ |2
|+𝐤
2
1
3
|
−1
Volume = |([2, 3, 1] × [1, −1, 2]) ⋅ [−1, 4, −4]| = |[7, −3, −5] ⋅ [−1, 4, −4]|
= |−7 − 12 + 20| = 1