Let u = 〈 u 1 , u 2 , 0 〉 and v = 〈 v 1 , v 2 , 0 〉 , be two-dimensional vector. The cross product of vectors u and v is net defined. However, if the vectors are regarded as the three-dimensional vectors u = 〈 u 1 , u 2 , 0 〉 and v = 〈 v 1 , v 2 , 0 〉 , respectively, then, in this case, we can define the cross product of u and v . In particular, in determinant notation, the cross product of u and v is given by u × v = | i j k u 1 u 2 0 v 1 v 2 0 | . Use this result to compute ( i cos θ + j sin θ ) × ( i sin θ − j cos θ ) , where θ is a real number.
Let u = 〈 u 1 , u 2 , 0 〉 and v = 〈 v 1 , v 2 , 0 〉 , be two-dimensional vector. The cross product of vectors u and v is net defined. However, if the vectors are regarded as the three-dimensional vectors u = 〈 u 1 , u 2 , 0 〉 and v = 〈 v 1 , v 2 , 0 〉 , respectively, then, in this case, we can define the cross product of u and v . In particular, in determinant notation, the cross product of u and v is given by u × v = | i j k u 1 u 2 0 v 1 v 2 0 | . Use this result to compute ( i cos θ + j sin θ ) × ( i sin θ − j cos θ ) , where θ is a real number.
Let
u
=
〈
u
1
,
u
2
,
0
〉
and
v
=
〈
v
1
,
v
2
,
0
〉
,
be two-dimensional vector. The cross product of vectors
u
and
v
is net defined. However, if the vectors are regarded as the three-dimensional vectors
u
=
〈
u
1
,
u
2
,
0
〉
and
v
=
〈
v
1
,
v
2
,
0
〉
,
respectively, then, in this case, we can define the cross product of
u
and
v
. In particular, in determinant notation, the cross product of
u
and
v
is given by
u
×
v
=
|
i
j
k
u
1
u
2
0
v
1
v
2
0
|
.
Use this result to compute
(
i
cos
θ
+
j
sin
θ
)
×
(
i
sin
θ
−
j
cos
θ
)
, where
θ
is a real number.
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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