Let u be a unit vector in the xy - plane of an xyz - coordinate system , and let v be a unit vector in the yz -plane. Let θ 1 be the angle between u and i, let θ 2 be the angle between v and k, and let θ be the angle between u and v. (a) Show that cos θ = ± sin θ 1 sin θ 2 . (b) Find θ if θ is acute and θ 1 = θ 2 = 45 ∘ . (c) Use a CAS to find, to the nearest degree, the maximum and minimum values of θ if θ is acute and θ 2 = 2 θ 1 .
Let u be a unit vector in the xy - plane of an xyz - coordinate system , and let v be a unit vector in the yz -plane. Let θ 1 be the angle between u and i, let θ 2 be the angle between v and k, and let θ be the angle between u and v. (a) Show that cos θ = ± sin θ 1 sin θ 2 . (b) Find θ if θ is acute and θ 1 = θ 2 = 45 ∘ . (c) Use a CAS to find, to the nearest degree, the maximum and minimum values of θ if θ is acute and θ 2 = 2 θ 1 .
Let u be a unit vector in the xy-plane of an xyz-coordinate system, and let v be a unit vector in the yz-plane. Let
θ
1
be the angle between u and i, let
θ
2
be the angle between v and k, and let
θ
be the angle between u and v.
(a) Show that
cos
θ
=
±
sin
θ
1
sin
θ
2
.
(b) Find
θ
if
θ
is
acute
and
θ
1
=
θ
2
=
45
∘
.
(c) Use a CAS to find, to the nearest degree, the maximum and minimum values of
θ
if
θ
is acute and
θ
2
=
2
θ
1
.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
Thomas' Calculus: Early Transcendentals (14th Edition)
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