How do you determine the unit normal vector?

To find the unit normal vector, first find the unit tangent vector. Finding the unit tangent vector requires differentiating each component of a vector function, as well as the length of the vector function. Finally, use the derivative of the unit tangent vector over the length of the unit tangent vector to determine the unit normal vector.

1. Differentiate the vector function, and set it to zero

   First, differentiate the given vector function: r(t) = cos ti + sin tj + tk. The derivative of this function is r'(t) = -sin ti + cos tj + k. The vector function is the point where t = 0 is: r'(0) = j + k.

2. Find the length of the vector function

   The length of the vector function is found by determining the square root of the square of each directional coefficient in the derivative: |r'(t)| = sqrt(1^2 + 1^2) = sqrt(2). Putting all of these elements together, the unit tangent vector is: T(t) = r'(t) / |r'(t)| = (1/sqrt(2)) * (-sin ti + cos tj + k).

3. Differentiate the unit tangent vector, and set it to zero

   Find the derivative of the unit tangent vector: T'(t) = (1/sqrt(2)) * (-cos ti - sin tj). Setting this to zero gives the result: T'(0) = -1/sqrt(2).

4. Find the length of the unit tangent vector

   For the unit tangent vector, the length is: |T'(t)| = (1/sqrt(2)).

5. Find the unit normal vector

   The unit normal vector is the unit tangent vector over the length of the unit tangent vector: N(t) = T'(t)/|T'(t)| = -cos ti - sin tj.