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
+ 2
x
3
= 4 it must have normal vector
n
=
(1
,
-
1
,
2). Thus the equation of the plane is
x
1
-
x
2
+ 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 simplified vector equation
which describes it.
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- Winter '08
- All
- Math, Linear Algebra, Vector Space, vector equation, ∈ Rn
-
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