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The tangent vector is a crucial concept when analyzing curves, as it represents the direction in which a curve is moving at any given point. For a curve described by the position vector \( \mathbf{r}(t) \), the tangent vector \( \mathbf{r}'(t) \) is the derivative of this position vector. The notation \( \frac{d}{dt} \) indicates differentiation with respect to the parameter \( t \).
For the given curve \( \mathbf{r}(t) = t \mathbf{i} + \ln(\cos t) \mathbf{j} \), its derivative, the tangent vector, is calculated as \( \mathbf{r}'(t) = \mathbf{i} - \tan(t) \mathbf{j} \).
This derivative gives the rate of change of the curve's position with time \( t \), essentially showing the direction of motion.

• **Vector Components:** Shows how changes in \( t \) affect each part (\( \mathbf{i} \) and \( \mathbf{j} \) components).
• **Slope Implication:** If \( \tan(t) \) varies, so does the curve's steepness or inclination.

Understanding the tangent vector helps comprehend the real-time dynamics of a curve.

A normal vector is orthogonal, or perpendicular, to a tangent vector at a given point on a curve. This means that while the tangent vector shows the direction of the curve, the normal vector highlights the direction pointing outward from the curve's path. This concept is crucial for understanding how a curve is bending at any point.
To find the normal vector \( \mathbf{N}(t) \), one must first derive the unit tangent vector's derivative. For the provided curve, after normalizing, the derivative of \( \mathbf{T}(t) = \cos(t) \mathbf{i} - \sin(t) \mathbf{j} \) is \( \mathbf{T}'(t) = -\sin(t) \mathbf{i} - \cos(t) \mathbf{j} \).

**Unit Normal Vector:**

• Calculated by normalizing \( \mathbf{T}'(t) \), the normal vector is expressed as \(-\sin(t) \mathbf{i} - \cos(t) \mathbf{j} \).
• Having a magnitude of 1 ensures it is a "unit" vector—perfectly adapted to express pure direction without respect to length.

The normal vector provides insights into the curve’s bending away from the tangent, serving as a foundation for understanding curvature.

A unit tangent vector, denoted \( \mathbf{T}(t) \), is a simplified version of the tangent vector. It has a length of 1. This makes it valuable for accurately representing direction without considering magnitude or length. To compute the unit tangent vector, divide the tangent vector by its magnitude, essentially normalizing it.
For example, from our calculation:

**Calculation Process:**

• Given \( \mathbf{r}'(t) = \mathbf{i} - \tan(t) \mathbf{j} \), its magnitude, or length, is \( \| \mathbf{r}'(t) \| = \sec(t) \).
• Then, the unit tangent vector is \( \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\| \mathbf{r}'(t) \|} = \cos(t) \mathbf{i} - \sin(t) \mathbf{j} \).

Using a unit tangent vector ensures a consistent and standardized way of assessing how the curve moves directionally, setting the stage for subsequent analyses like finding normal vectors and understanding curvature.