WEEK ONE HW SOLUTIONS
Let
A
=
a
1
i
+
a
2
j
+
a
3
k
And remember the formula for projection:
proj
B
A
=
°
A
·
B
B
·
B
±
B
1.
What are the projections of
A
onto the standard basis vectors
i
,
j
, and
k
? How do you interpret
this result?
Answer.
proj
i
A
=
°
A
·
i
i
·
i
±
i
=(
A
·
i
)
i
=(
a
1
+ 0 + 0)
i
=
a
1
i
Similarly proj
j
A
=
a
2
j
and proj
k
A
=
a
3
k
The projection of any vector onto the standard basis is the standard decomposition into the
distance along the xaxis, yaxis and zaxis.
2.
You have decided that you are more of a cylindrical person you think outside the box. In other
words, you want to express
A
in terms of a diFerent set of basis vectors and have decided to use
the following vectors:
•
a unit vector
e
r
that points in the direction
a
1
i
+
a
2
j
+0
k
;
•
a unit vector
e
θ
that points in the direction
−
a
2
i
+
a
1
j
+0
k
;
•
a unit vector
e
z
=
k
.
Determine expressions for
e
r
and
e
θ
in terms of
a
1
,
a
2
, and
a
3
, and the standard basis vectors.
Remember that
e
r
,
e
θ
and
e
z
must be unit vectors. What can you say about the orthogonality
of these new basis vectors? Write
A
in terms of the three new basis vectors (Hint: think about
projecting
A
onto the new basis vectors.) How do you interpret the projection of
A
onto each of
these three new unit vectors?
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 Fall '07
 ADAMNORRIS
 Linear Algebra, Standard basis, F3, basis vectors, new basis vectors

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