(1) Let
~v
=
h
7
,
5
i
and
(yellow/pink)
~w
=
h
4
,

3
i
(green)
=
h
6
,

3
i
(blue)
=
h
8
,

3
i
. Use these values for all questions on this page.
(a) Compute 6

10
.
(yellow/pink)
6
h
7
,
5
i
10
h
4
,

3
i
=
h
42
,
30
ih
40
,

30
i
=
h
42

40
,
30

(

30)
i
=
h
2
,
60
i
(green)
6
h
7
,
5
i
10
h
6
,

3
i
=
h
42
,
30
ih
60
,

30
i
=
h
42

60
,
30

(

30)
i
=
h
18
,
60
i
(blue)
6
h
7
,
5
i
10
h
8
,

3
i
=
h
42
,
30
ih
80
,

30
i
=
h
42

80
,
30

(

30)
i
=
h
38
,
60
i
(b) Find all unit vectors perpendicular to
.
The vectors perpendicular to
h
a, b
i
are all the scalar multiples of
h
b,

a
i
.In
this case,
h
5
,

7
i
is perpendicular to
h
7
,
5
i
:
h
5
,

7
i·h
7
,
5
i
= 5(7)

7(5) = 0.
The length is
h
5
,

7
i
=
p
5
2
+(

7)
2
=
√
25+49=
√
74.
One unit vector is
h
5
,

7
i
/
5
,

7
=
D
5
/
√
74
,

7
/
√
74
E
The other one is its negative
D

5
/
√
74
,
7
/
√
74
E
.
(c) Compute

10 proj
9ˆ
ı
(
).
(yellow/pink)
proj
~a
(
~
b
)=
(
~
a
·
~
b
)
~
a

~
a

2
; here,
=9ˆ
ı
,
~
b
=
~
w
proj
(
~
b
(9ˆ
ı
·h
4
,

3
i
)9ˆ
ı
9
2
=
(9(4)+0(

3))9ˆ
ı
9
2
=
(9
2
)(4)ˆ
ı
9
2
=4ˆ
ı
.