A review for the first test
A summary of the most useful formulas of Chapt 12
For a curve of equation ~r(t) = (x(t), y(t), z(t))
Z
t
1. (arc length) s(t1 ) =
||r0 (t)||dt
t0
2. (tangent line at ~r(t0 )) (x, y, z) = r~0 (t0 )(t − t0 ) + ~r(t0 )
3. (unit tangent vector at t) T~ (t) = r0 (t)/||r0 (t)||
r~0 (t) × r~00 (t)
~
4. (binormal unit vector at t) B(t)
=
||r~0 (t) × r~00 (t)||
~ (t) = B(t)
~ × T~ (t)
5. (normal unit vector at t) N
6. (tangential vector component of the acceleration) ~aT (t) = (~a(t).T~ (t))T~ (t)
7. (normal vector component of the acceleration) ~aN (t) = ~a(t) − ~aT (t)
8. (normal scalar component of the acceleration)
9. (curvature at ~r(t)) k(t) =
dT~ (t)
/||r~0 (t)||
dt
10. (radius of curvature at ~r(t)) ρ(t) =
=
||~aN (t)|| = aN (t) =
||r~0 (t)×r~00 (t)||
||r~0 (t)||
||r~0 (t)×r~00 (t)||
||r~0 (t)||3
1
k(t)
A list of proofs (possible extra credit problems).
• the formula for the dot product of vectors,
• the formula for the cross product of vectors,
• the formula for the area of a parallelogram and the volume of a parallelepiped
•
dT~ (t)
dt
is perpendicular to T (t)
Problems
1. For which values of the parameter λ are the vectors v = (2, 2, 3) and w = ( λ2 , 3, 2λ) a)
perpendicular, b) form an obtuse angle, c) parallel?
2. find two unit vector perpendicular to w = −2i + 4j.
3. find the component of the projection projw (v) of the vector v = −i + 3j + 6k over the
vector w and c) find the work done by a force v applied to a point that moves a distance
of 15 m. over a ramp parallel to w.
4. Let u = 3i − 3k , v = 2i − 3j + k and w = i − 4j. Find a) the area of the parallelogram
that has u and v as adjacent sides and b) the volume of the parallelepiped that has u,
v and w as adjacent sides.
5. a) Find the parametric equation of a line s that passes through (0, 1, 0) and (3, 0, 1).
b) Find the equation of a line r that is parallel to s and passes through (0, 1, −2).
d) Find the equation of a plane that contains both r and s
e) Find (if possible) the intersection of the line s in a) and the line s0 that is parallel to
(1, 2, 3) and passes through (6, 1, 2)
6. a) Find the equation of a plane β that is perpendicular to the line l(t) = (2t + 1)i + (2t −
1)j + k and passes through the origin. At which point do the plane and s intersect?
b) Find the line of intersection of β and the plane 3x − y + z = 0
√ 3
3
7. a) Find the arc length of the curve r(t) = (t 2 , t, 23 2 t 2 ), t ∈ [0, 1] and its arc length
parametrization
8. a) Find the equation of the tangent line to the curve r(t) = 3(sin t)2 i + 2(cos t)2 j +
3(ln t)2 k at t = π, and b) unit tangent vector at this point.
9. Find the vectors T(t), N(t) and B(t) for the curve r(t) = t sin ti + 2t cos tj + 3tk at
t = 0.
10. Find the velocity, acceleration and speed of a particle that moves along the curve r(t) =
e2t i + 2e−2t j + tk. Find also the minimum speed of the particle and the position where
this occurs.
11. A particle moves along a curve; its velocity is described by the equation
v(t) = i + 2 cos tj + 2 sin tk.
a) Find the position function of the particle, assuming that it start its motion from the
point P0 = (0, 1, 0).
b) Find the displacement of the particle and the distance travelled by the particle for
t ∈ [0, 1]
c) Find the acceleration of the particle and its tangential and normal components.
12. Find a formula for the curvature of the curve r(t) = (t2 , t3 , 2t) at a generic point t.
Then use this formula to find the radius of curvature of the curve at t = 0.
p
1 − x2 − y 2 − 1
p
13. Find the domain of the functions a) f (x, y) =
and b) g(x, y) =
x2 + y 2
log(1 − x2 − y) − 1
p
. Sketch a picture of your domain marking clearly the areas where
x2 + y 2
the function is undefined.