Determine whether the statement about the vector field F( x , y ) is true or false. If false. explain why. F( x , y ) = x x 2 + y 2 i − y x 2 + y 2 j . (a) As x , y moves away from the origin, the lengths of the vectors decrease. (b) If x , y is a point on the positive x - axis, then the vector points up. (c) If x , y is a point on the positive y - axis , the vector points to the right.
Determine whether the statement about the vector field F( x , y ) is true or false. If false. explain why. F( x , y ) = x x 2 + y 2 i − y x 2 + y 2 j . (a) As x , y moves away from the origin, the lengths of the vectors decrease. (b) If x , y is a point on the positive x - axis, then the vector points up. (c) If x , y is a point on the positive y - axis , the vector points to the right.
Determine whether the statement about the vector field
F(
x
,
y
)
is true or false. If false. explain why.
F(
x
,
y
)
=
x
x
2
+
y
2
i
−
y
x
2
+
y
2
j
.
(a) As
x
,
y
moves away from the origin, the lengths of the vectors decrease.
(b) If
x
,
y
is a point on the positive
x
-
axis,
then the vector points up.
(c) If
x
,
y
is a point on the positive
y
-
axis
,
the vector points to the right.
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.
Sketch the graph of the vector-valued function r(t) = (2t – 1)² î + (2t +2) ĵ.
Draw arrows on your graph to indicate the orientation.
Assume that an object is moving along a parametric curve and the three vector function.
T (t), N(t), and B (t) all exist at a particular point on that curve.
CIRCLE the ONE statement below that MUST BE TRUE:
(a) B. T=1
(b) T x B = N (B is the binormal vector.)
v (t)
(c) N (t) =
|v (t)|
(d) N (t) always points in the direction of velocity v (t).
(e) a (t) lies in the same plane as T (t) and N (t).
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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