Question

Consider the helix given by ⟨acos(t),asin(∣t),b(t)⟩ with a,b⟩0

A.) Stetch the graph, label t=0,t=π/2,t=−π/2 and the motion as t→+[infinity]
B.) Find the uniat Tangent vector
C-1 Find the unit normal vector
D.) Find the curvature
E.) Find the unit binormal

Answer 1

To stretch the graph, label key points and describe motion. Find the unit Tangent and Normal vectors, curvature, and unit Binormal vector.

Step-by-step explanation:

To stretch the graph, we need to consider the values of t at certain intervals. For t = 0, the helix begins at its starting point. For t = π/2, the helix reaches the highest point in the positive y-direction. For t = -π/2, the helix reaches the highest point in the negative y-direction. As t approaches positive infinity, the helix continues to spiral upwards.

The unit Tangent vector, T, can be found by taking the derivative of the helix equation and normalizing it. The unit Normal vector, N, can be found by taking the derivative of the unit Tangent vector and normalizing it. The curvature, k, can be found by taking the magnitude of the derivative of the unit Tangent vector. The unit Binormal vector, B, can be found by taking the cross product of the unit Tangent and unit Normal vectors.

Unit Tangent vector (T): T = <-sin(t), cos(t), b'(t)>

Unit Normal vector (N): N = <-cos(t), -sin(t), b''(t)>

Curvature (k): k = |b''(t)|

Unit Binormal vector (B): B = T × N