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When you think about a tangent vector, picture it as the arrow that just touches the curve at a specific point, pointing in the direction the curve is headed. This vector tells you the immediate direction in which the curve is moving.

- **Find the Derivative:** The first step in finding the tangent vector is to take the derivative of the position vector, \( \mathbf{r}(t) \). This gives you \( \mathbf{r}'(t) \), which is essentially the velocity of the object moving along the curve.- **Normalization:** To convert this tangent vector into a unit tangent vector, \( \mathbf{T} \), you normalize it by dividing by its magnitude. This step helps us understand the direction of the tangent without being concerned about its length.For the given curve, \( \mathbf{r}'(t) = (-t \cos t, t \sin t) \), and once normalized, we have \( \mathbf{T} = (-\cos t, \sin t) \). This means our tangent vector points exactly in the direction defined by the angle \( t \).

A normal vector is perpendicular to the tangent vector and points towards the inside of the curve's bend. It's like the curve's imaginary 'compass needle,' always at a right angle to where you're heading instantaneously.

- **Differentiate the Tangent Vector:** The normal vector originates from taking the derivative of the unit tangent vector \( \mathbf{T}(t) \). This derivative gives you the change in direction of the tangent vector.- **Normalization Again:** Like the tangent vector, the normal vector \( \mathbf{N} \) also needs to be a unit vector, ensuring it has a length of one unit.For our curve, taking the derivative yields \( \mathbf{T}'(t) = (\sin t, \cos t) \), and it happens to already be a unit vector: \( \mathbf{N} = (\sin t, \cos t) \). This normal vector gives us an insight into the orientation and the curvature of our path.

Curvature is like the measure of how sharply a curve bends at a given point. It’s akin to asking, 'How tight is the curve here?' Higher curvature means a tighter bend and vice versa.

- **Formula for Curvature:** For curves parameterized by \( t \), curvature \( \kappa \) is calculated using the derivative of the tangent vector and the derivative of the position vector.- **Calculation:** In our example, we find \( \kappa \) using the formula \( \kappa = \frac{| \mathbf{T}'(t) |}{| \mathbf{r}'(t) |} \), which gives us \( \kappa = \frac{1}{|t|} \).This concise measure allows you to understand how much the path turns at any point \( t \). Curvature becomes extremely useful in fields like physics and engineering, where knowing how a path bends can be vital to design.