[Collect homework #2]
[Collect summaries of section]
[Hand out questions for diagnostic quiz #2]
Why is it that if
a
•
(
b
×
c
) = 0, then
a
,
b
, and
c
are
coplanar?
..?..
b
×
c
must be perpendicular to
a
, and since
b
×
c
is also
perpendicular to
b
and
c
, the vectors
a
,
b
, and
c
are all
perpendicular to
b
×
c
;
so
a
,
b
, and
c
must be coplanar (specifically, they lie in the
plane perpendicular to
b
×
c
).
This analysis presupposes that
b
×
c
is not the zero vector.
How do we prove the claim in the case where
b
×
c
is the
zero vector?
..?..
b
and
c
and parallel (they lie on a line), so
a
,
b
, and
c
lie in
a plane.
Conversely, if
a
,
b
, and
c
are coplanar,
a
•
(
b
×
c
) must be
0, as we’ll now see.
The triple product:
a
•
(
b
×
c
) = the signed volume
V
of the parallelepiped
spanned by
a
,
b
, and
c
(which vanishes if
a
,
b
, and
c
are
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coplanar).
More precisely, the triple product equals +
V
or –
V
,
according to whether …
..?..
a
,
b
, and
c
form a righthanded triple or a lefthanded triple.
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 Fall '11
 Staff
 Vectors, Vector Space, Euclidean vector, Bivector

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