Finding the Unit Tangent Vector
To find the unit tangent vector
t
at a specific point defined by a parameter t using the given parametric equations, follow these steps:
Step 1: Identify the Parametric Equations
The curve is defined by the position vector:
r(t) = (t^2, 2t, 1 + 3t + 13t^3 + 12t^2)
Step 2: Calculate the Derivative
The first step is to compute the derivative of the position vector, r(t), with respect to t. This will give us the velocity vector:
r'(t) = (2t, 2, 3 + 39t^2 + 24t)
Step 3: Evaluate the Derivative at the Given Parameter
Next, substitute the given value of t into the derivative to find the corresponding velocity vector:
r'(t_0) = (2t_0, 2, 3 + 39t_0^2 + 24t_0)
Step 4: Compute the Magnitude of the Velocity Vector
To find the unit tangent vector, we need the magnitude of the velocity vector:
|r'(t_0)| =
√((2t_0)^2 + 2^2 + (3 + 39t_0^2 + 24t_0)^2)
Step 5: Find the Unit Tangent Vector
Finally, the unit tangent vector t can be computed by dividing the velocity vector by its magnitude:
t(t_0) =
\frac{r'(t_0)}{|r'(t_0)|} = \frac{(2t_0, 2, 3 + 39t_0^2 + 24t_0)}{|r'(t_0)|}
And that’s how you find the unit tangent vector at a given point!