Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 18]_The Vector Function of a Scalar Argument_237
2078. Show that the vector equation rr, = (/* 2 r,) /,
where r l and r 2 are radius vectors of two given points, is the
equation of a straight line.
2079. Determine which lines are hodographs of the following
vector functions:
a) r = at -f c\ c) r = a cos t -f b sin t\
b) r = a/ 2
+ ft/; d) r = a cosh / -f 6 sinh /,
where a, ft, and c are constant vectors; the vectors a and b
are perpendicular to each other.
2080. Find the derivative vector-function of the function
a(t) = a(t)a (/), where a(t) is a scalar function, while a(/)
is a unit vector, for cases when the vector a(t) varies: 1) in
length only, 2) in direction only, 3) in length and in direction
(general
2081.
case). Interpret geometrically the results obtained.
Using the rules of differentiating a vector functisn with
respect to a scalar argument, derive a formula for differentiating
a mixed product of three vector functions a, 6, and c.
of
2082. Find the derivative, with respect to
the volume of a parallelepiped constructed
the parameter t,
on three vectors:
2083. The equation
a = i + tj+t z
k,
of motion is
r = 3/cos/~j-4/sinf,
where / is the time. Determine the trajectory of motion, the
velocity and the acceleration. Construct the trajectory of motion
and the vectors of velocity and acceleration for times, / = 0,
.
2084. The equation of motion is
Determine the trajectory of motion, the velocity and the acceleration.
What are the magnitudes of velocity and acceleration
and what directions have they for time = and / = -y?
2085. The equation of motion is
r= i cos a cos at -\-jsmacostot + k sincof,
where a and o are constants and / is the time. Determine the
trajectory of motion and the magnitudes and directions o! the
velocity and the acceleration.