For Problems 19 − 21 , determine the inner product of the given vectors using (a) the inner product ( 5.1.14 ) in Problem 18 , (b) the standard inner product in ℝ 2 . v = ( 1 , − 2 ) , w = ( 2 , 1 ) . 18. Consider the vector space ℝ 2 . Define the mapping 〈 , 〉 by 〈 v , w 〉 = 2 v 1 w 1 + v 1 w 2 + v 2 w 1 + 2 v 2 w 2 ( 5.1.14 ) for all vectors v = ( v 1 , v 2 ) and w = ( w 1 , w 2 ) in ℝ 2 . Verify that Equation ( 5.1.14 ) defines an inner product on ℝ 2 .
For Problems 19 − 21 , determine the inner product of the given vectors using (a) the inner product ( 5.1.14 ) in Problem 18 , (b) the standard inner product in ℝ 2 . v = ( 1 , − 2 ) , w = ( 2 , 1 ) . 18. Consider the vector space ℝ 2 . Define the mapping 〈 , 〉 by 〈 v , w 〉 = 2 v 1 w 1 + v 1 w 2 + v 2 w 1 + 2 v 2 w 2 ( 5.1.14 ) for all vectors v = ( v 1 , v 2 ) and w = ( w 1 , w 2 ) in ℝ 2 . Verify that Equation ( 5.1.14 ) defines an inner product on ℝ 2 .
For Problems
19
−
21
, determine the inner product of the given vectors using (a) the inner product
(
5.1.14
)
in Problem
18
, (b) the standard inner product in
ℝ
2
.
v
=
(
1
,
−
2
)
,
w
=
(
2
,
1
)
.
18. Consider the vector space
ℝ
2
. Define the mapping
〈
,
〉
by
〈
v
,
w
〉
=
2
v
1
w
1
+
v
1
w
2
+
v
2
w
1
+
2
v
2
w
2
(
5.1.14
)
for all vectors
v
=
(
v
1
,
v
2
)
and
w
=
(
w
1
,
w
2
)
in
ℝ
2
. Verify that Equation
(
5.1.14
)
defines an inner product on
ℝ
2
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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