Let F be the vector field 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 field 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 expression displaystyle 'int' is positive, negative, or zero. The line segment is in the direction of path of the vectors
(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.
A net is dipped in a river. Determine the
flow rate of water across the net if the
velocity vector field for the river is given
by v=(x-y,z+y+7,z2) and the net is
decribed by the equation y=1-x2-z2, y20,
and oriented in the positive y- direction.
(Use symbolic notation and fractions
where needed.)
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.
A fire ant, searching for hot sauce in a picnic area, goes through three displacements along level ground: d→1 for 0.41 m southwest (that is, at 45° from directly south and from directly west), d→2 for 0.52 m due east, and d→3 for 0.77 m at 60° north of east. Let the positive x direction be east and the positive y direction be north. What are (a) the x component and (b) the y component of d→1? What are (c) the x component and (d) the y component of d→2? What are (e) the x component and (f) the y component of d→3? What are (g) the x component and (h) the y component, (i) the magnitude, and (j) the direction of the ant's net displacement? If the ant is to return directly to the starting point, (k) how far and (l) in what direction should it move? Give all angles as positive (counterclockwise) angles relative to the +x-axis.
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