Question

Consider the following vector function: r(t)=<5t, t²/2, t²>

a) Find the unit tangent and unit normal vectors T(t) and N(t).

Answer 1

To find the unit tangent and unit normal vectors, first find the velocity vector by taking the derivative of the position vector. Then normalize the velocity vector to get the unit tangent vector. Finally, take the derivative of the unit tangent vector to get the unit normal vector.

Step-by-step explanation:

To find the unit tangent and unit normal vectors, we need to first find the velocity vector and then normalize it to get the unit tangent vector. The velocity vector is obtained by taking the derivative of the position vector. So, let's find the derivative of the vector function r(t) = <5t, t²/2, t²>:

r'(t) = <5, t, 2t>

Now, to find the unit tangent vector, divide the velocity vector by its magnitude:

T(t) = (1/√(5² + t² + 4t²)) <5, t, 2t>

The unit normal vector, N(t), is obtained by taking the derivative of the unit tangent vector:

N(t) = T'(t) = <0, (5-4t)/√(5² + t² + 4t²), -2t/√(5² + t² + 4t²)>