Question

consider the vector function given below.r(t) = 6t, 5 cos t, 5 sin tfind the unit tangent and unit normal vectors t(t) and n(t).

Answer 1

To find the unit tangent and unit normal vectors t(t) and n(t) for the given vector function r(t), differentiate the function to find the velocity and acceleration vectors, then divide those vectors by their magnitudes to find the unit tangent and unit normal vectors.

Step-by-step explanation:

To find the unit tangent and unit normal vectors t(t) and n(t), we need to find the velocity vector and acceleration vector first.

Given the vector function r(t) = 6t, 5cos(t), 5sin(t), we can differentiate it with respect to time to find the velocity vector:

v(t) = 6, -5sin(t), 5cos(t)

Next, we differentiate the velocity vector to find the acceleration vector:

a(t) = 0, -5cos(t), -5sin(t)

To find the unit tangent vector t(t), we divide the velocity vector by its magnitude:

t(t) = (v(t))/|v(t)| = (6, -5sin(t), 5cos(t))/sqrt(6^2 + (-5sin(t))^2 + (5cos(t))^2)

To find the unit normal vector n(t), we divide the acceleration vector by its magnitude:

n(t) = (a(t))/|a(t)| = (0, -5cos(t), -5sin(t))/sqrt((-5cos(t))^2 + (-5sin(t))^2)