The tetrahedron defined by three vectors v → 1 , v → 2 , v → 3 in ℝ 3 is the set of all vectors of the form c 1 v → 1 + c 2 v → 2 + c 3 v → 3 ,where c i ≥ 0 and c 1 + c 2 + c 3 ≤ 1 . Explain why thevolume of this tetrahedron is one sixth of the volume ofthe parallelepiped defined by v → 1 , v → 2 , v → 3 .
The tetrahedron defined by three vectors v → 1 , v → 2 , v → 3 in ℝ 3 is the set of all vectors of the form c 1 v → 1 + c 2 v → 2 + c 3 v → 3 ,where c i ≥ 0 and c 1 + c 2 + c 3 ≤ 1 . Explain why thevolume of this tetrahedron is one sixth of the volume ofthe parallelepiped defined by v → 1 , v → 2 , v → 3 .
Solution Summary: The author explains that the tetrahedron's volume is one sixth of the parallelepiped defined by the given vector.
The tetrahedron defined by three vectors
v
→
1
,
v
→
2
,
v
→
3
in
ℝ
3
is the set of all vectors of the form
c
1
v
→
1
+
c
2
v
→
2
+
c
3
v
→
3
,where
c
i
≥
0
and
c
1
+
c
2
+
c
3
≤
1
. Explain why thevolume of this tetrahedron is one sixth of the volume ofthe parallelepiped defined by
v
→
1
,
v
→
2
,
v
→
3
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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