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Math 136
Assignment 1 Solutions
1.
Compute each of the following.
a) (1
,
3
,
4) + (

1
,
1
,
2)
Solution: (1
,
3
,
4) + (

1
,
1
,
2) = (0
,
4
,
6).
b) 3(

1
,
1
,

2)

2(2
,
0
,
3).
Solution: 3(

1
,
1
,

2)

2(2
,
0
,
3) = (

3
,
3
,

6) + (

4
,
0
,

6) = (

7
,
3
,

12).
2.
Determine the distance between
P
(2
,
1
,
1) and
Q
(1
,

1
,
1).
Solution: The distance is
±
(1
,

1
,
1)

(2
,
1
,
1)
±
=
±
(

1
,

2
,
0)
±
=
±
(

1)
2
+(

2)
2
+0
2
=
√
5.
3.
Determine which of the following pairs of vectors is orthogonal.
a) (2
,
1
,
1), (

2
,
1
,
3).
Solution: We have (2
,
1
,
1)
·
(

2
,
1
,
3) =

4 + 1 + 3 = 0 so they are orthogonal.
b) (

1
,
3
,
6), (3
,

1
,
0).
Solution: We have (

1
,
3
,
6)
·
(3
,

1
,
0) =

3

3 + 0 =

6 so they are not orthogonal.
4.
Find an equation for the plane through
P
(3
,

2
,
1) and parallel to
x
1

x
2
+2
x
3
= 4.
Solution: Since the plane is parallel to
x
1

x
2
x
3
= 4 it must have normal vector
±n
=
(1
,

1
,
2). Thus the equation of the plane is
x
1

x
2
x
3
= 1(3)

1(

2) + 2(1) = 7
.
5.
For each of the following sets:
i) Determine if the set is linearly dependent or linearly independent. Justify.
ii) Describe geometrically the span of the set and give a simpli±ed vector equation
which describes it.
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This note was uploaded on 04/30/2010 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.
 Winter '08
 All
 Math

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