If v 1 , v 2 , and v 3 are noncoplanar vectors , let k 1 = v 2 × v 3 v 1 ⋅ ( v 2 × v 3 ) k 2 = v 3 × v 1 v 1 ⋅ ( v 2 × v 3 ) k 3 = v 1 × v 2 v 1 ⋅ ( v 2 × v 3 ) (These vectors occur in the study of crystallography. Vectors of the from n 1 v 1 + n 2 v 2 + n 3 v 3 , where each n i is an integer, from a lattice for a crystal. Vectors written similarly in terms of k 1 , k 2 , and k 3 from the reciprocal lattice .) (a) Show that k i is perpendicular to v j if i ≠ j . (b) Show that k i ⋅ v i = 1 for i = 1 , 2 , 3 . (c) Show that k 1 ⋅ ( k 2 × k 3 ) = 1 v 1 ⋅ ( v 2 × v 3 ) .
If v 1 , v 2 , and v 3 are noncoplanar vectors , let k 1 = v 2 × v 3 v 1 ⋅ ( v 2 × v 3 ) k 2 = v 3 × v 1 v 1 ⋅ ( v 2 × v 3 ) k 3 = v 1 × v 2 v 1 ⋅ ( v 2 × v 3 ) (These vectors occur in the study of crystallography. Vectors of the from n 1 v 1 + n 2 v 2 + n 3 v 3 , where each n i is an integer, from a lattice for a crystal. Vectors written similarly in terms of k 1 , k 2 , and k 3 from the reciprocal lattice .) (a) Show that k i is perpendicular to v j if i ≠ j . (b) Show that k i ⋅ v i = 1 for i = 1 , 2 , 3 . (c) Show that k 1 ⋅ ( k 2 × k 3 ) = 1 v 1 ⋅ ( v 2 × v 3 ) .
Solution Summary: The author explains that atimes b is orthogonal to both vectors, i.e.
If
v
1
,
v
2
, and
v
3
are noncoplanar vectors, let
k
1
=
v
2
×
v
3
v
1
⋅
(
v
2
×
v
3
)
k
2
=
v
3
×
v
1
v
1
⋅
(
v
2
×
v
3
)
k
3
=
v
1
×
v
2
v
1
⋅
(
v
2
×
v
3
)
(These vectors occur in the study of crystallography. Vectors of the from
n
1
v
1
+
n
2
v
2
+
n
3
v
3
, where each
n
i
is an integer, from a lattice for a crystal. Vectors written similarly in terms of
k
1
,
k
2
, and
k
3
from the reciprocal lattice.)
(a) Show that
k
i
is perpendicular to
v
j
if
i
≠
j
.
(b) Show that
k
i
⋅
v
i
=
1
for
i
=
1
,
2
,
3
.
(c) Show that
k
1
⋅
(
k
2
×
k
3
)
=
1
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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