Consider unit vector v → 1 , ... , v → n in R n such that the angle between v → i and v → j is 60 ° for all i ≠ j . Find the n -volume of the n -parallelepiped spanned by v → 1 , ... , v → n . Hint : Let A = [ v → 1 ⋅ ⋅ ⋅ v → n ] , and think about the matrix A T A and its determinant. Exercise 17 is useful.
Consider unit vector v → 1 , ... , v → n in R n such that the angle between v → i and v → j is 60 ° for all i ≠ j . Find the n -volume of the n -parallelepiped spanned by v → 1 , ... , v → n . Hint : Let A = [ v → 1 ⋅ ⋅ ⋅ v → n ] , and think about the matrix A T A and its determinant. Exercise 17 is useful.
Solution Summary: The author explains that the determinant of a matrix of the form set is the volume of our n-parallelepiped.
Consider unit vector
v
→
1
,
...
,
v
→
n
in
R
n
such that the angle between
v
→
i
and
v
→
j
is
60
°
for all
i
≠
j
. Find the n-volume of the n-parallelepiped spanned by
v
→
1
,
...
,
v
→
n
. Hint: Let
A
=
[
v
→
1
⋅
⋅
⋅
v
→
n
]
, and think about the matrix
A
T
A
and its determinant. Exercise 17 is useful.
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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