Partial Solution Set, Leon Section 5.1
5.1.1c
Find the angle between
v
= (4
,
1)
T
and
w
= (3
,
2)
T
.
Solution
: We have
v
T
w
= 14,

v

=
√
17, and

w

=
√
13, so the angle between
v
and
w
is
θ
= arccos
14
√
221
.
5.1.2c
Using the same vectors as in the preceding problem, the vector projection of
v
onto
w
is
14
13
w
= (42
/
13
,
28
/
13)
T
. The vector projection of
w
onto
v
is
14
17
v
= (56
/
17
,
14
/
17)
T
.
5.1.3
For each pair of vectors
x
and
y
, compute the vector projection
p
of
x
onto
y
, and
verify that
p
is orthogonal to
x

p
.
(b)
x
= (3
,
5)
T
,
y
= (1
,
1)
T
(d)
x
= (2
,

5
,
4)
T
,
y
= (1
,
2
,

1)
T
.
Solution
:
(b) The vector projection is
p
=
x
T
y
y
T
y
y
= 4
y
= (4
,
4)
T
. Clearly
p
is orthogonal to
x

p
= (

1
,
1)
T
since their dot product is zero.
(d) The vector projection is
p
=
x
T
y
y
T
y
y
=

2
y
= (

2
,

4
,
2)
T
. As in (b), the veriﬁca
tion of orthogonality is trivial.
5.1.5
Find the point on the line
y
= 2
x
that is closest to the point (5
,
2).
Solution
: We want to ﬁnd the vector projection of
v
= (5
,
2)
T
onto some vector
w
that
is colinear with the given line. Any choice will do, say
w
= (1
,
2)
T
. The projection is
v
T
w
w
T
w
w
=
9
5
w
= (9
/
5
,
18
/
5)
T
.
5.1.7
Find the distance from the point (1
,
2) to the line 4
x

3
y
= 0.