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.
A vector F represents a force that has a magnitude of 8 lb and π/3 is the radian measure of its direction angle. Find the work done by the force in moving an object
(a) along the x -axis from the origin to the point (6, 0) and
(b) along the y-axis from the origin to the point (0, 6). Distance is measured in feet.
Find (u+v).(2u-v) given that u.u=4, u.v=-5 and v.v=10Determine if u and v are orthogonal u=(cosθ,sinθ,-1); v=(sinθ,-cosθ,0)
Determine the magnitude of the vector difference V' = V₂ - V₁ and the angle 0x which V' makes with the positive x-axis. Complete both
(a) graphical and (b) algebraic solutions.
Assume a = 3, b = 5, V₁ = 8 units, V₂ = 14 units, and 0 = 51°
V₂
Answers:
(a) V' = i
(b) ex = i
-x
units
Thomas' Calculus: Early Transcendentals (14th Edition)
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