Suppose x= (2t+1)/(t-1) ,y= (t^2)/(t-1) , z=(t+2). Determine the unit tangent ,the principal normal, curvature , radius of curvature, the binormal , torsion , radius of torsion
Answers
Answer:
Step-by-step explanation:
Solution: u =
−−−→ P1P2 = h2, 3, 0i, v =
−−−→ P1P3 = h1, −1, 1i, w = h1, 4, −1i, and the scalar triple
product is equal to
u • (v × w) =
2 3 0
1 −1 1
1 4 −1
= 2
−1 1
4 −1
− 3
1 1
1 −1
+ 0
1 −1
1 4
= 0,
so the volume of the parallelepiped determined by u, v, and w is equal to 0. This means that
these vectors are on the same plane. So, P1, P2, P3, and P4 are coplanar.
2. (10 points) Find the equation of the plane that is equidistant from the points A = (3, 2, 1)
and B = (−3, −2, −1) (that is, every point on the plane has the same distance from the two
given points).
Solution: The midpoint of the points A and B is the point C =
1
2
3, 2, 1)+(−3, −2, −1)
=
(0, 0, 0). A normal vector to the plane is given by −−→AB = h3, 2, 1i − h−3, −2, −1i = h6, 4, 2i.
So, the equation of the plane is 6(x − 0) + 4(y − 0) + 2(z − 0) = 0, that is, 3x + 2y + z = 0.
3. (6 points) Find the vector projection of b onto a if a = h4, 2, 0i and b = h1, 1, 1i.
Solution: Since |a|
2 = 42 + 22 = 20, the vector projection of b onto a is is equal to
projab =
a • b
|a|
2
a =
h4, 2, 0i • h1, 1, 1i
20
=
6
20
h4, 2, 0i =
3
5
h2, 1, 0i.
4. (12 points) Consider the curve r(t) = √
2 costi + sin tj + sin tk.
(a) (8 points) Find the unit tangent vector function T(t) and the unit normal vector function
N(t).
(b) (4 points) Compute the curvature κ.
Solution: (a) r
0
(t) = −
√
2 sin t i + costj + costk and |r
0
(t)| =
p
2 sin2
t + cos2 t + cos2
√
t =
2. So, the unit tangent vector T(t) is is equal to
T(t) = r
0
(t)
|r
0(t)|
= − sin ti +
√
2
2
costj +
√
2
2
costk.
Since T0
(t) = − costi −
√
2
2
sin tj −
√
2
2
sin tk and |T0
(t)| =
q
cos2 t +
1
2
sin2
t +
1
2
sin2
t = 1,
the normal vector is equal to
N(t) = costi −
√
2
2
sin tj −
√
2
2
sin tk.