Let F be the vector fields shown in the figure. (a) If C 1 is the vertical line segment from ( − 3 , − 3 ) to ( − 3 , 3 ) , determine whether ∫ C 1 F ⋅ d r is positive, negative, or zero. (b) If C 2 is the counterclockwise-oriented circle with radius 3 and center the origin, determine whether ∫ C 2 F ⋅ d r is positive, negative, or zero.
Let F be the vector fields shown in the figure. (a) If C 1 is the vertical line segment from ( − 3 , − 3 ) to ( − 3 , 3 ) , determine whether ∫ C 1 F ⋅ d r is positive, negative, or zero. (b) If C 2 is the counterclockwise-oriented circle with radius 3 and center the origin, determine whether ∫ C 2 F ⋅ d r is positive, negative, or zero.
Solution Summary: The author explains that the line integral of F along C is positive.
(a) If C1 is the vertical line segment from
(
−
3
,
−
3
)
to
(
−
3
,
3
)
, determine whether
∫
C
1
F
⋅
d
r
is positive, negative, or zero.
(b) If C2 is the counterclockwise-oriented circle with radius 3 and center the origin, determine whether
∫
C
2
F
⋅
d
r
is positive, negative, or zero.
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.
You are on a rollercoaster, and the path of your body is modeled by a vector function r(t),
where t is in seconds, the units of distance are in feet, and t = 0 represents the start of the
ride. Assume the axes represent the standard cardinal directions and elevation (x is E/W, y
is N/S, z is height). Explain what the following would represent physically, being as specific
as possible. These are all common roller coaster shapes/behaviors and can be explained in
specific language with regard to units:
a. r(0)=r(120)
b. For 0 ≤ t ≤ 30, N(t) = 0
c. r'(30) = 120
d. For 60 ≤ t ≤ 64, k(t) =
40
and z is constant.
e.
For 100 ≤ t ≤ 102, your B begins by pointing toward positive z, and does one full
rotation in the normal (NB) plane while your T remains constant.
Find a vector in component form that is perpendicular to the line 3x 4y = 6.
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