1.
Math1151 Algebra. S1 2010 Test 2. Version 2a.
(1) Is the vector
u
= (1
,

5
,

2)
T
parallel to the plane
x
= (2
,
4
,

1)
T
+
λv
+
μw
where
v
= (1
,
2
,
2)
T
and
w
= (3
,

1
,
7)
T
?
Solution.
The vector
u
is parallel to the plane above if
u
∈
span
{
v, w
}
, that is, if
there exists
λ, μ
such that
v
=
λv
+
μw
. Putting into matrix form and row
reducing gives
1
3

1
2

1


5
2
7


2
→
1
3

1
0

7


7
0
1


4
→
1
3

1
0

7


7
0
0


5
so
u
=
λv
+
μw
have no solutions. Hence
u
∈
span
{
v, w
}
.
(2) Compute
D
=
1
3

1
2
1
0

1
5
7
Solution.
D
=
1(7)

3(14)

(10 + 1) =

46
(3) Find the shortest distance from the point with coordinate vector
a
=
(3
,

1
,
4)
T
to the plane Π with equation
x
+ 3
y
+ 3
z
= 5.
Solution.
We first pick a point
b
= (

1
,
1
,
1) on Π and note that
n
= (1
,
3
,
3) is
normal to the Π. The shortest distance from
a
to Π is given by
a

b
= (4
,

2
,
3)
d
=

proj
n
(
a

b
)

=

(
a

b
)
·
n


n

=
7
√
19
.
(4) A fifth of the items handled by a courier company are parcels, and the
rest are documents.
The company delivers 80% of parcels on time and
delivers 90% of documents on time. If an item is selected at random, find
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