Jan 30, 2008
Math 20C
Winter 2008
M
IDTERM
1(
A
)
Solutions
1
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View Full Document1. The following three space vectors are all parallel to the same plane which is this very
sheet of paper:
±w
±v
±u
where the vectors
±v
and
±w
are parallel to each other.
For problems (i) – (vi) below, state whether the given object is:
(a) the number zero,
(b) the zero vector,
(c) a nonzero vector perpendicular to the page,
(d) a nonzero vector parallel to the page,
(e) none of the above or undeﬁned.
No justiﬁcation required. Each choice might appear more than once.
(i)
proj
±w
±v
(d).
The projection of any vector onto a vector
±w
is always either a vector parallel
to
±w
, or the zero vector. Since
±v
is neither zero nor perpendicular to
±w
, its projection
to
±w
must be parallel to
±w
, in particular to this page.
(ii)
(
±w
×
±u
)
·
±v
(a).
±w
×
±u
is perpendicular to the page because
±u
and
±w
are parallel to the page and
not to each other. In particular,
±w
×
±u
is perpendicular to
±v
.
(iii)
(
±v
·
±w
)
±u
(d).
±v
·
±w
is a nonzero number, for the two vectors are not perpendicular to each
other. So this is a nonzero scalar multiple of
±u
, which is parallel to the page.
(iv)
±u
×
(
±u
×
±v
)
(d).
±u
×
±v
is perpenicular to the page, and
±u
×
(
±u
×
±v
)
is perpendicular to
±u
×
±v
.
The answer follows from the fact that any vector perpendicular to a normal vector
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 Winter '08
 Helton
 Vectors, Vector Space, normal vector

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