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

Sec. 19] The Natural Trihedron of a Space Curve 241
Hence, the osculating plane is defined by
or
{dx,
dx t 0} and
|o,
the vectors
- yd* 2 ,
{1, 1, 0} and {0, 1, 1}.
Whence the normal vector of the osculating plane is
and, therefore, its equation is
that is,
J
B= 1 1
-1 1
-l(x-l
= lj k
jdx*\
as it should be, since our curve is located in this plane.
2090. Find the basic unit vectors T, v, p of the curve
x^l cosf, y=sin/, z = t
at the point f = -g- *
2091. Find the unit vectors of the tangent and the principal
normal of the conic spiral
at an arbitrary point. Determine the angles that these lines make
with the z-axis.
2092. Find the basic unit vectors r, v, p of the curve
at the point x = 2.
y x*, z = 2x
2093. For the screw line
y = asmt, z = bt
write the equations of the straight lines that form a natural
trihedron at an arbitrary point of the line. Determine the direction
cosines of the tangent line and the principal normal.
2094. Write the equations of the planes that form the natural
trihedron of the curve
2
x* -1- 1/ + * 2 = 6, x 2
if -1- z 2 - 4
at one of its points M(l, 1, 2).
2095. Form the equations ot the tangent line, the normal
plane and the osculating plane of the curve * = /, y = t*, z = t*
at the point M (2, 4, 8).