Two unit vectors , a 1 and a 2, lie in the xy plane and pass through the origin. They make angles ∅ 1 and ∅ 2 , respectively, with the x axis (a) Express each vector in rectangular components; (b) take the dot product and verify the trigonometric identity, cos ( ϕ 1 − ϕ 2 ) = cos ϕ 1 cos ϕ 2 + sin ϕ 1 sin ϕ 2 ; (c) take the cross product and verify the trigonometric identity sin ( ϕ 2 − ϕ 1 ) = sin ϕ 2 cos ϕ 1 − cos ϕ 2 sin ϕ 1 .
Two unit vectors , a 1 and a 2, lie in the xy plane and pass through the origin. They make angles ∅ 1 and ∅ 2 , respectively, with the x axis (a) Express each vector in rectangular components; (b) take the dot product and verify the trigonometric identity, cos ( ϕ 1 − ϕ 2 ) = cos ϕ 1 cos ϕ 2 + sin ϕ 1 sin ϕ 2 ; (c) take the cross product and verify the trigonometric identity sin ( ϕ 2 − ϕ 1 ) = sin ϕ 2 cos ϕ 1 − cos ϕ 2 sin ϕ 1 .
Two unit vectors, a1 and a2, lie in the xy plane and pass through the origin. They make angles ∅1 and ∅2, respectively, with the x axis (a) Express each vector in rectangular components; (b) take the dot product and verify the trigonometric identity,
cos
(
ϕ
1
−
ϕ
2
)
=
cos
ϕ
1
cos
ϕ
2
+
sin
ϕ
1
sin
ϕ
2
; (c) take the cross product and verify the trigonometric identity
sin
(
ϕ
2
−
ϕ
1
)
=
sin
ϕ
2
cos
ϕ
1
−
cos
ϕ
2
sin
ϕ
1
.
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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