Find the equations of the tangent line, normal plane and osculating plane to the curve
r(t) = -2sin(t) i + 2cos(t) j + 3 k
at the point corresponding to t = π/4.
T[/B]^(t) = r'(t) // ||r'(t)||
u = a i + b j + c k, ||u|| = √(a^2 + b^2 + c^2)
N^(t) = T^(t) / ||T^(t)||
B^ = T^ × N^
The osculating plane is plane formed by unit normal and unit tangent vectors where the unit binormal vector is the vector normal to the osculating plane.
The normal plane is formed by the unit normal and unit binormal vectors and the unit tangent vector is the vector normal to the normal plane.
The Attempt at a Solution
N[/B]^(t) = sin(t) i - cos(t) j
B(t) = k (technically independent of t)
This was a question on a quiz I received, and I'm studying for an exam now, and I have the following three questions.:
- Would the tangent line be represented by P= a ⋅ T^(π/4) + [r(t) - r(π/4)] (where a is some scalar)?
- Would the normal plane be represented by T^(pi/4) ⋅ (r(t) -r(π/4)) = 0?
- Would the osculating plane be represented by B^(π/4) ⋅ (r(t) - r(π/4)) = 0?
Any input would be GREATLY appreciated!