8.
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SCalcET7 12.4.053.
Suppose that
a
≠
0
(a) If
a
·
b
=
a
·
c
, does it follow that
b
.
=
c
?

(b) If
a
×
b
=
a
×
c
, does it follow that
b
=
c
?

(c) If
a
·
b
=
a
·
c
, and
a
×
b
=
a
×
c
, does it follow that
b
=
c
?
Solution or Explanation
(a) No. If
a
·
b
=
a
·
c
, then
a
· (
b
−
c
) = 0, so
a
is perpendicular to
b
−
c
, which can happen if
b
≠
c
. For example, let
and

(b) No. If
a
×
b
=
a
×
c
, then
a
× (
b
−
c
) =
0
, which implies that
a
is parallel to
b
−
c
, which of
course can happen if
b
≠
c
.
(c) Yes. Since
a
·
c
=
a
·
b
,
a
is perpendicular to
b
−
c
, by part (a). From part (b),
a
parallel to
b
−
c
. Thus since
a
≠
0
but is both parallel and perpendicular to
b
−
c
, we have
b
−
c
=
0
, so
b
=
is also
c
.